\documentclass[ALCO, Unicode]{cedram}

\usepackage{mathtools}
\usepackage{tikz}
\usetikzlibrary{patterns}

\usepackage{caption}
\usepackage{subcaption}

\newcommand\note[1]{\textbf{\color{red}#1}}

\newcommand\Defn[1]{\textbf{\color{black}#1}}
\newcommand\Def[1]{\Defn{#1}}

\renewcommand\u{\mathbf{u}}
\renewcommand\v{\mathbf{v}}
\newcommand\w{\mathbf{w}}
\newcommand\p{\mathbf{p}}
\newcommand\q{\mathbf{q}}

\newcommand\polar{\vee}

\newcommand\eps{\varepsilon}
\renewcommand\emptyset{\varnothing}
\newcommand\Z{\mathbb{Z}}
\newcommand\Znn{\mathbb{Z}_{\ge0}}
\newcommand\N{\mathbb{N}}
\newcommand\R{\mathbb{R}}
\newcommand\Rnn{\R_{\ge0}}
\newcommand\C{\mathbb{C}}
\newcommand\z{\mathbf{z}}
\newcommand\x{\mathbf{x}}
\newcommand\y{\mathbf{y}}
\newcommand\s{\mathbf{s}}
\newcommand\inner[1]{\langle {#1} \rangle}
\newcommand\defeq{\coloneqq}
\newcommand\eqdef{\eqqcolon}

\newcommand\F{\mathcal{F}}
\renewcommand\a{\alpha}
\newcommand\AInt{\widehat{A}}
\newcommand\AExt{\widecheck{A}}
\newcommand\CAb{\beta}
\newcommand\bInt{\widehat{\CAb}}
\newcommand\bExt{\widecheck{\CAb}}
\newcommand\Sphere{\mathbb{S}}

\newcommand\op{\mathrm{op}}

\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\relint}{relint}
\DeclareMathOperator{\interior}{int}
\DeclareMathOperator{\aff}{aff}
\newcommand\affL{\aff_0}

\DeclareMathOperator{\cpl}{cpl}

\DeclareMathOperator{\lineal}{lineal}
\DeclareMathOperator{\conv}{conv}
\DeclareMathOperator{\cone}{cone}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\rk}{rk}

\DeclareMathOperator{\abR}{R}
\DeclareMathOperator{\abE}{E}
\DeclareMathOperator{\abP}{Pr}
\DeclareMathOperator{\abM}{M}

\DeclareMathOperator{\Ima}{Im}

\DeclareFontFamily{U}{mathx}{\hyphenchar\font45}
\DeclareFontShape{U}{mathx}{m}{n}{ <5> <6> <7> <8> <9> <10> <10.95> <12> <14.4>
  <17.28> <20.74> <24.88> mathx10 }{}
\DeclareSymbolFont{mathx}{U}{mathx}{m}{n}
\DeclareFontSubstitution{U}{mathx}{m}{n}
\DeclareMathAccent{\widecheck}{0}{mathx}{"71}

\theoremstyle{plain}
\newtheorem{quest}{Question}

\theoremstyle{remark}
\newtheorem{obs}{Observation}
\newtheorem{rmk}{Remark}

\title{Generalized angle vectors, geometric lattices, and flag-angles}

\author{\firstname{Spencer} \lastname{Backman}}
\address{Department of Mathematics and Statistics, University of Vermont, Burlington, Vermont, USA}
\email{Spencer.Backman@uvm.edu}

\author{\firstname{Sebastian} \lastname{Manecke}}
\address{Institut f\"ur Mathematik, Goethe-Universit\"at Frankfurt, Germany}
\email{manecke@math.uni-frankfurt.de}

\author{\firstname{Raman} \lastname{Sanyal}}
\address{Institut f\"ur Mathematik, Goethe-Universit\"at Frankfurt, Germany}
\email{sanyal@math.uni-frankfurt.de}


\keywords{interior/exterior angles, cone valuations, Gram's relation,
graded posets, angle deficiencies, flag-angles, flag-vectors, incidence
algebras}

\subjclass[2020]{52B11, 52B45, 52C35, 06A11, 52B12}

\begin{document}

\begin{abstract}
    Interior and exterior angle vectors of polytopes capture curvature
    information at faces of all dimensions and can be seen as metric
    variants of $f$-vectors. In this context, Gram's relation takes the place of the
    Euler--Poincar\'e relation as the unique linear relation among interior
    angles. We show the existence and uniqueness of Euler--Poincar\'e-type
    relations for generalized angle vectors
    by building a bridge to the
    algebraic combinatorics of geometric lattices, generalizing work of
    Klivans--Swartz.

    We introduce flag-angles of polytopes as a geometric counterpart to flag-$f$-vectors.
    Flag-angles generalize the angle deficiencies of Descartes--Shephard, Grassmann angles, and
    spherical intrinsic volumes. Using the machinery of incidence algebras, we relate flag-angles
    of zonotopes to flag-$f$-vectors of graded posets. This allows us to determine the linear
    relations satisfied by interior/exterior flag-angle vectors.
\end{abstract}
\maketitle


\newcommand\Tcone[2]{\mathrm{T}_{#1}{#2}}
\newcommand\SCA{\nu}
\newcommand\intSCA{\widehat{\SCA}}

\newcommand\Ncone[2]{\mathrm{N}_{#1}#2}
\newcommand\extSCA{\widecheck{\SCA}}

\newcommand\CA{\alpha}
\newcommand\aInt{\widehat{\CA}}
\newcommand\aExt{\widecheck{\CA}}

\newcommand\Cones{\mathcal{C}}
\newcommand\LL{\mathsf{L}}
\newcommand\Lflats{\mathcal{L}}
\newcommand\Ltop{\mathbf{1}}
\newcommand\Lbot{\mathbf{0}}

\newcommand{\Incidence}{\mathcal{I}}
\newcommand\Arr{\mathcal{H}}
\newcommand\Lfaces{\mathcal{F}}

\newcommand\poset{\mathcal{P}}
\newcommand\FP{\mathcal{F}}
\newcommand\FW{\mathbf{W}}


\newcommand\faInt{\widehat{\boldsymbol{\CA}}}
\newcommand\faExt{\widecheck{\boldsymbol{\CA}}}

\section{Introduction}\label{sec:intro}
For a convex polytope $P \subset \R^d$ of dimension $d$, let $f_i(P)$ be the number of
\mbox{$i$-dimensional} faces of $P$ for $i = 0,1,\dots,d-1$. The Euler--Poincar\'e relation states
that the face numbers satisfy
\begin{equation}\label{eqn:euler}
    f_0(P) - f_1(P) + f_2(P) - \cdots + (-1)^{d-1} f_{d-1}(P) \ = \ 1 -
    (-1)^d \, .
\end{equation}
This simple linear relation among the face numbers is the key to a rich interplay of geometry,
combinatorics, and algebra as amply demonstrated, for example, in~\cite{RotaKlain,
Stanley-eulerian}. In particular, the Euler--Poincar\'e relation~\eqref{eqn:euler} is, up to
scaling, the only linear relation among the face numbers of polytopes of fixed dimension $d$;
cf.~\cite{hohn},~\cite[Sect.~8.1]{gruenbaum}. In this paper, we will show the existence and
uniqueness of Euler--Poincar\'e-type relations for certain class of \emph{semi-discrete} invariants
and their interplay with algebraic combinatorics. These invariants are generalizations of the
well-known \emph{interior} and \emph{exterior} angle vectors of polytopes.

The local geometry of $P$ at a face $F$ is captured by the \Def{tangent cone} (or \Def{inner cone})
$\Tcone{F}{P} \defeq \cone(-F + P)$, which is the cone of feasible directions from $F$. The
\Def{interior angle} of $P$ at $F$ is the spherical volume
\[
    \intSCA(F,P) \ \defeq \ \SCA(\Tcone{F}{P}) \ = \
    \frac{\vol(\Tcone{F}{P} \cap B_d)}{\vol(B_d)} \, ,
\]
which generalizes the notion of dihedral angle to faces of all dimensions. The $i$-th interior
  angle of $P$ is given by $\intSCA_i(P) \defeq \sum_F \intSCA(F,P)$, where $F$ ranges over all
  faces of dimension $i$. The \Def{interior angle vector} $\intSCA(P) =
  (\intSCA_i(P))_{i=0,\dots,d-1}$ captures curvature information at faces of various dimensions and
  combines combinatorial as well as metric properties of $P$. The fundamental relation, called
  \Def{Gram's relation}, which is satisfied by interior angle vectors of $d$-dimensional polytopes
  is
\begin{equation}\label{eqn:SCAgram}
        \intSCA_0(P) - \intSCA_1(P) + \intSCA_2(P) - \cdots + (-1)^{d-1}
        \intSCA_{d-1}(P) \ = \ (-1)^{d+1} \, .
\end{equation}
We refer to Gr\"unbaum~\cite[Sect.~14.4]{gruenbaum} for a detailed historic account of Gram's
relation and its importance.

Gromov and Milman~\cite{GM87} introduced and studied an \emph{anisotropic} notion of angle of a
cone $C \subset \R^d$
\[
    \SCA_K(C) \ \defeq \ \frac{\vol(C \cap K)}{\vol(K)} \, ,
\]
where $K \subset \R^d$ is a fixed centrally-symmetric convex body. These \emph{cone (probability)
  measures} are related to surface measures of general (star) convex bodies~\cite{NR03,
  Naor07,BGMN05}. Using the integral-geometric perspective developed by
  Perles--Shephard~\cite{PS67} (see also~\cite{Welzl}), one easily shows that Gram's relation is
  the essentially unique linear relation for the associated interior angles $\intSCA_{K}(P)_i =
  \sum_F \SCA_K(\Tcone{F}{P})$ of $d$-dimensional polytopes.

In the first part of the paper, we prove the existence and uniqueness of Gram's relation for a more
general class of cone angles by way of an astounding connection to the combinatorics of posets and
matroids.

A map $\CA : \Cones_d \to R$ from the collection of convex polyhedral cones $\Cones_d$ in $\R^d$
into some unital ring $R$ is a \Def{valuation} if $\CA(\{0\}) = 0$ and
\[
    \CA(C \cup C') \ = \ \CA(C) + \CA(C') - \CA(C \cap C') \,
\]
for all $C,C' \in \Cones_d$ such that $C \cup C', C \cap C' \in \Cones_d$. A valuation $\CA$ is
  \Def{simple} if $\CA(C) = 0$ whenever $\dim C < d$ and we call $\CA$ a \Def{cone angle} if in
  addition $\CA(\R^d) = 1$. Cone valuations play a decisive role in integral
  geometry~\cite{SchneiderWeil} and cone angles strictly subsume cone probability measures; see
  Section~\ref{sec:CA}. Note that we do not require cone angles to be rotationally invariant or to
  satisfy any positivity conditions. The \Def{interior $\CA$-angle} of a polytope $P$ at a face $F
  \subseteq P$ is then defined as $\aInt(F,P) \defeq \CA(\Tcone{F}{P})$. We also define the
  \Def{exterior $\CA$-angle} of $P$ at $F$ as $\aExt(F,P) \defeq \CA(\Ncone{F}{P} + \affL(F))$,
  where $\affL(F)$ is the linear subspace parallel to $F$ and $\Ncone{F}{P}$ is the \Def{normal
  cone} of $P$ at $F$, that is, the cone polar to $\Tcone{F}{P}$. As expected, the interior and
  exterior \Def{$\CA$-angle vector} of $P$ are defined through
\[
    \aInt_i(P) \ \defeq \ \sum_{F} \aInt(F,P) \qquad \text{ and } \qquad
    \aExt_i(P) \ \defeq \ \sum_{F} \aExt(F,P) \, ,
\]
where the sums are over all faces $F \subseteq P$ of dimension $i$, for $i = 0, \dots, d-1$. Our
  first main result is this.

\begin{theo}\label{thm:gram}
    Let $\CA$ be a cone angle. Then, up to scaling, the only linear
    relations satisfied by $\aInt(P)$, respectively $\aExt(P)$, for
    any $d$-dimensional polytope $P$ are
    \begin{equation} \label{eqn:gram} 
        \aInt_0(P) - \aInt_1(P) + \aInt_2(P) - \cdots + (-1)^{d-1}
        \aInt_{d-1}(P) \ = \ (-1)^{d+1} \, ,
    \end{equation}
    \begin{equation}
      \label{eqn:ext_gram}
        \aExt_0(P) \ = \ \sum_{v} \aExt(v,P) \ = \ 1 \,.
    \end{equation}
\end{theo}
Showing the validity of both relations is not difficult. Indeed,
in~\cite{PS67} a proof of~\eqref{eqn:SCAgram} is sketched that works for
general cone angles. For completeness, we give a proof using a conical version
of the Brianchon--Gram relation of~\cite{AS15}; see
also~\cite{lawrence,shephard-elem}. The main challenge in proving
Theorem~\ref{thm:gram} is uniqueness, as none of the analytic and geometric
properties of $\SCA_K$ carry over to general cone angles.  We prove
Theorem~\ref{thm:gram} by establishing a powerful connection between the
geometry and the combinatorics of zonotopes.

To explain the combinatorial connection, define $\LL(F) \defeq \affL(F)^\perp$ for any non-empty
face $F \subseteq P$ and let $\Lflats(P) \defeq \{\LL(F) : \emptyset \neq F \subseteq P \}$
partially ordered by \emph{reverse} inclusion. This is a finite graded poset of rank $d$. In
particular, if $P$ is a zonotope, that is, a Minkowski-sum of segments, then $\Lflats(P)$ is a
geometric lattice, called \Defn{lattice of flats}. The \Def{Whitney numbers} of the first kind
$w_i$ and of the second kind $W_i$ of a graded poset are important enumerative
invariants~\cite[Sect.~3.10]{EC1}, whose precise definition we recall in Section~\ref{sec:zono}.
The following result allows us to show the uniqueness in Theorem~\ref{thm:gram} on a purely
combinatorial level.

\begin{theo}\label{thm:whitney}
    Let $Z \subset \R^d$ be a $d$-dimensional zonotope with lattice of flats
    $\Lflats = \Lflats(Z)$. For any cone angle $\CA$ we have
    \[
        \aExt_i(Z) \ = \ W_{i}(\Lflats(Z)) \quad \text{ and } \quad
        \aInt_i(Z) \ = \ (-1)^{d-i} w_{d-i}(\Lflats(Z)^\op)
    \]
    for all $i=0,\dots,d-1$.
\end{theo}

For the standard cone angle, the second equation in Theorem~\ref{thm:whitney} was proven in
Klivans--Swartz~\cite{KlivansSwartz} using ideas similar to those in~\cite{PS67} involving
projections of zonotopes. Theorem~\ref{thm:whitney} is then used to show that interior/exterior
angle vectors of zonotopes certify the uniqueness of~\eqref{eqn:gram} and~\eqref{eqn:ext_gram}.
Theorem~\ref{thm:whitney} and Theorem~\ref{thm:gram} are proved in Section~\ref{sec:zono}.

In Section~\ref{sec:incalg}, we recast the correspondence between interior/exterior angles and
Whitney numbers of the first and second kind in algebraic terms. McMullen~\cite{McM-angle} showed
that $\aInt,\aExt$ can be interpreted as elements in the incidence algebra $\Incidence(\Lfaces(P))$
of the face poset $\Lfaces(P)$ of $P$. In Section~\ref{sec:incalg}, we show that for a zonotope $Z$
with lattice of flats $\Lflats$, there is a subalgebra $\Incidence_\LL \subseteq
\Incidence(\Lfaces(Z))$ such that the map $F \mapsto \LL(F)$ yields a ring map $ \LL_* :
\Incidence_\LL \to \Incidence(\Lflats)$. We show that $\aExt \in \Incidence_\LL$ and $\LL_* \aExt =
\zeta_{\Incidence(\Lflats)}$. McMullen's \emph{inverse} angles~\cite{McM-polytopealgebra} then
allow us to show $\LL_* \aInt' = \mu_{\Incidence(\Lflats)}$, where $\aInt'$ is a slight
modification of $\aInt$. This yields an elegant algebraic proof of Theorem~\ref{thm:whitney} and
explains the appearance of Whitney numbers of the first and second kind. In addition, we give a
simple proof of a beautiful relation due to Klivans and Swartz~\cite{KlivansSwartz} between
spherical intrinsic volumes of a zonotope $Z$ and the characteristic polynomial of $\Lflats(Z)$;
see Corollary~\ref{cor:alg-KS}.

The second goal of the paper is to introduce \emph{flag-angle vectors} as a unifying geometric
concept and to exhibit and exploit parallels to the theory of flag-vectors of posets. To motivate
flag-angle vectors, let $P$ be a $3$-dimensional polytope. Descartes defined the \emph{angle
defect} at a vertex $v$ as $\delta(v,P) = 1 - \sum_F \intSCA(v,F)$, where the sum is over all
$2$-dimensional faces $F$ containing $v$ and he showed that
\[
    \sum_v \delta(v,P) \ = \ 2 \, .
\]
For a $d$-polytope $P$ and $i=0,\dots,d-3$, Shephard~\cite{Shephard-angle} defines the
  \emph{$i$-th total angle deficiency}
\[
    \delta_i(P) \ \defeq \ \ f_i(P) - \sum_{G \subset F} \intSCA(G,F) \, ,
\]
where the sum is over all faces $G \subset F$, with $\dim G = i$ and $\dim F = d-1$. Generalizing
  Descartes' result, Shephard showed that
\begin{equation}\label{eqn:angle_defect}
    \delta_0(P) - \delta_1(P) + \dots + (-1)^{d-3} \delta_{d-3}(P) \ = \ 1 +
    (-1)^{d-1}\,.
\end{equation}
Such generalized Descartes-relations were further studied for manifolds~\cite{GrunbaumShephard} and
in relation with stratified curvature~\cite{Bloch} and polyhedral Gauss--Bonnet
theorems~\cite{Schneider-GaussBonnet}.

In contrast to interior angle vectors, total angle deficiencies record the interaction of faces of
various dimensions and the generalized Descartes--relation~\eqref{eqn:angle_defect} shows that
angle deficiencies are not independent. In a different direction, McMullen~\cite{McM-angle} showed
that exterior angles can be computed from interior angles of flags of faces
\begin{equation}\label{eqn:int_to_ext}
    (-1)^d \widecheck{\SCA}_i(P) \ = \ \sum_{F_1 \subset F_2 \subset \cdots
    \subset F_k} (-1)^{k+1} 
    \intSCA(F_1,F_2) 
    \intSCA(F_2,F_3) \cdots
    \intSCA(F_{k},P)  \, ,
\end{equation}
where the sum is over all flags of faces with $\dim F_1 = i$. Moreover, McMullen showed that
various other measures of curvature, such as spherical intrinsic volumes and Gr\"{u}nbaum's
\emph{Grassmann angles}~\cite{Grunbaum-grassmann} can be computed from chains of interior (or
exterior) angles.

\begin{defi}
    Let $\CA$ be a cone angle. For a $d$-dimensional polytope $P$ and a
    non-empty set $S = \{ 0 \le s_1 < s_2 < \cdots < s_k \le d-1 \}$, define the
    \Def{interior flag-angle} by
    \begin{equation}\label{eqn:flag}
        \aInt_S(P) \ \defeq \ \sum_{F_1 \subset F_2 \subset \cdots \subset F_k}
        \aInt(F_1,F_2) \,  \aInt(F_2,F_3) \cdots \aInt(F_k,P) \, ,
    \end{equation}
    where the sum is over all chains of faces of $P$ such that $\dim F_i = s_i$ for $i=1,\dots,k$.
    The \Def{exterior flag-angle} $\aExt_S(P)$ is defined analogously and we set
    $\aExt_\emptyset(P) \defeq \aInt_\emptyset(P) \defeq 1$.
\end{defi}

The vectors $\faInt(P) = (\aInt_S(P))_S$ and $\faExt(P) = (\aExt_S(P))_S$ are called the interior
and exterior \Def{flag-angle vectors} of $P$.

In Sections~\ref{sec:flag} and~\ref{sec:flag-whitney}, we determine the affine spaces spanned by
interior and exterior flag-angle vectors, respectively.

\begin{theo}\label{thm:flag_rels}
    Let $P$ be a $d$-dimensional polytope and $S \subseteq [d-1] \defeq \{1, 2, \dots, d-1\}$.
    For any cone angle $\CA$, we have
    \[
        \aExt_{S}(P) \ = \ \aExt_{S \cup \{0\}}(P)
        \qquad \text{ and } \qquad
        \sum_{i=0}^{t-1} (-1)^{i} \aInt_{S \cup \{i\}}(P) \ = \ (-1)^{t+1}
        \aInt_S(P) \, ,
    \]
    where $t = \min(S \cup \{d\})$.

    Moreover, the affine hull of exterior flag-angles as well as the affine hull of interior
    flag-angles is of dimension $2^{d-1} - 1$. These spaces are spanned by the flag-angle vectors
    of zonotopes.
\end{theo}

Since $\delta_i(P) = f_i(P) - 2 \intSCA_{i,d-1}(P)$, Theorem~\ref{thm:flag_rels} for $S = \{d-1\}$
together with the Euler--Poincar\'e relation~\eqref{eqn:euler} implies Shephard's
result~\eqref{eqn:angle_defect}. Theorem~\ref{thm:flag_rels} also determines the spaces of linear
relations on all specializations of flag-angles, including spherical intrinsic volumes and
Grassmann angles.

Flag angle vectors are \emph{semi-discrete} counterparts to \emph{flag vectors} of polytopes and
posets. Let $\poset$ be a graded poset of rank $d+1$ with minimal element $\Lbot$ and maximal
element $\Ltop$. For $S \subseteq [d]$, the number of chains of elements
\[
    \Lbot \ \prec_\poset \ c_1 \ \prec_\poset \ c_2 \ \prec_\poset \ \cdots \
    \prec_\poset \ c_k \ \prec_\poset \ \Ltop
\]
such that $S = \{ \rk c_1, \rk c_2, \dots, \rk c_k \}$ is called the \Def{flag-Whitney number} of
  the \Def{second kind} $W_S(\poset)$. The resulting vector $\FW(\poset)=(W_S(\poset))_{S}$ is
  commonly known as the \Def{flag-vector} of $\poset$. Flag-vectors of face posets of polytopes or,
  more generally, of Eulerian posets have received considerable attention, starting with the
  seminal paper Bayer--Billera~\cite{BB}. Bayer and Billera determined the linear relations on
  flag-vectors of Eulerian posets, called the \emph{generalized Dehn-Sommerville relations} and
  showed that flag-vectors of Eulerian posets of rank $d+1$ span an affine space of dimension
  $F_d$, where $F_d$ is the $d$-th Fibonacci number. Billera--Ehrenborg--Readdy~\cite{BER} showed
  that these spaces are spanned by the flag vectors of $d$-dimensional zonotopes.

To complete the relation to flag vectors, we introduce the \Defn{flag-Whitney numbers} of the
\Def{first kind} $w_S(\poset)$, which extend the ordinary Whitney numbers $w_i(\poset)$ to flags.
They are obtained via inclusion-exclusion from the flag-vector and are complementary to the usual
\emph{flag $h$-vector}. The following result extends Theorem~\ref{thm:whitney} to flag-angle
vectors. We set $d - S \defeq \{ d - s : s \in S \}$.

\begin{theo}\label{thm:flag_whitney}
    Let $\CA$ be a cone angle and $Z \subset \R^d$ a full-dimensional
    zonotope. Then for a nonempty $S \subseteq \{0,1,\dots,d-1\}$, we have
    \[
        \aExt_S(Z) \ = \ W_{S}(\Lflats(Z)) \quad \text{ and } \quad
        \aInt_S(Z) \ = \ (-1)^{d-r} w_{d - S}(\Lflats(Z)^\op) \, ,
    \]
    where $r = \min(S)$.
\end{theo}

We prove Theorems~\ref{thm:flag_rels} and~\ref{thm:flag_whitney} in Section~\ref{sec:flag} using
the algebraic tools developed in Section~\ref{sec:incalg}. As before,
Theorem~\ref{thm:flag_whitney} is the key to proving the main statement of
Theorem~\ref{thm:flag_rels}. Corollary~\ref{cor:int_to_ext} extends McMullen's
relation~\eqref{eqn:int_to_ext} to flag-angles, which then shows that it suffices to only consider
exterior flag-angle vectors. This amounts to showing that there are no linear relations on
flag-vectors of $\Lflats(Z)$ for $d$-dimensional zonotopes $Z$, which we do in
Theorem~\ref{thm:zono_span}. This strengthens a result of Billera--Hetyei~\cite{BilleraHetyei}.
Since the flag-vector of a zonotope is encoded by the flag-vector of its lattice of flats by
results in~\cite{BayerSturmfels}, Theorem~\ref{thm:zono_span} also strengthens the main result in
Billera--Ehrenborg--Readdy~\cite{BER} in that flag-vectors of Eulerian posets are spanned by the
flag vectors of zonotopes. In Section~\ref{sec:spherical_intrinsic_volumes} we revisit
Grassmann-angle and spherical intrinsic volumes from the perspective of Crofton-type formulas and
introduce a generalization for cone angles. This allows us to generalize the main result of
Gr\"unbaum's fundamental paper~\cite{Grunbaum-grassmann} on Grassmann angles of polytopes.

\section{Cone angles and linear relations}\label{sec:CA}

In this section, we recall the basic geometric constructions and prove the existence of the
relations stated in Theorem~\ref{thm:gram}.

Let $P \subset \R^d$ be a full-dimensional polytope and $q \in P$. The \Def{tangent cone} of $P$ at
$q$ is
\[
    \Tcone{q}{P} \ \defeq \{ u \in \R^d : q + \eps u \in P \text{ for some }
    \eps > 0 \} \ = \ \cone(-q + P) \, .
\]
It is easy to verify that $\Tcone{q}{P} = \Tcone{q'}{P}$ if and only if $q,q' \in \relint(F)$ for
  some face $F \subseteq P$ and we recover $\Tcone{F}{P} = \Tcone{q}{P} = \cone(-F + P)$. Tangent
  cones capture the local geometry of $P$ at $F$ in that faces of $\Tcone{F}{P}$ are in
  correspondence to faces $G \supseteq F$ under the correspondence $G \mapsto \Tcone{F}{G}$. If $P$
  is not full-dimensional, we extend the definition to
\[
    \Tcone{F}{P} \ = \ \cone(-F + P) + \affL(P)^\perp \, .
\]

\newcommand\Ocone[2]{\mathrm{O}_{#1}#2}
The \Def{normal cone} of a face $F \subseteq P$ is the polyhedral cone
\[
    \Ncone{F}{P} \ \defeq \ \{ c \in \R^d \ : \ \inner{c,x} \ge \inner{c,y}
    \text{ for all } x \in F, y \in P \} \, .
\]
By construction, $\Ncone{F}{P}$ is contained in $\affL(F)^\perp$. We define the \Def{outer cone}
  of $P$ at $F$ as the $d$-dimensional cone
\[
    \Ocone{F}{P} \ \defeq \ \Ncone{F}{P} + \affL(F) \, .
\]

As stated in the introduction, a cone angle $\CA : \Cones_d \to R$ is a simple valuation normalized
so that $\CA(\R^d)=1$. Cone probability measures $\SCA_K(C) = \frac{\vol(C \cap K)}{\vol(K)}$ for
$K$ a full-dimensional convex body are cone angles, but the notion of cone angles is richer. For
example, for any point $q \in \R^d$, let $B_\eps(q)$ be the ball with radius $\epsilon > 0$
centered at $q$. Then
\[
    \omega_q(C) \ \defeq \ \lim_{\eps \to 0} \frac{\vol(B_\eps(q) \cap
        C)}{\vol(B_\eps(q))}
\]
defines a cone angle which, for $q \neq 0$ does not come from a measure. Further instances can be
  obtained, for example, from the construction of Dehn--Hadwiger
  functionals~\cite[Sect.~2.2.2]{hadwiger}.

The following standard construction yields a \emph{universal} cone valuation. Let $\Z\Cones_d$ be
the free abelian group with generators $e_C$ for $C \in \Cones_d$. Let $\mathcal{U} \subset
\Z\Cones_d$ be the subgroup generated by
\[
    e_{C \cup D} + e_{C \cap D} - e_{C} - e_{D}
\]
\newcommand\CG{\mathbb{S}} for all cones $C,D \in \Cones_d$ such that $C \cup D, C \cap D \in \Cones_d$
and let $\mathcal{S}$ be the subgroup generated by all elements $e_C$ for
which $\dim C < d$. We call $\CG \defeq \Z\Cones_d / (\mathcal{U} +
\mathcal{S})$ the \Def{simple cone group}. If $h$ is a homomorphism
from $\CG$ to some abelian group $G$, then $\phi_h : \Cones_d \to G$ with
$\phi_h(C) \defeq h(e_C)$ is a simple valuation.
Volland~\cite{volland} essentially showed that every valuation lifts to
a homomorphism on $\CG$. We record this as follows.
\begin{theo}[Volland]
    The map $\Cones_d \to \CG$ given by $C \mapsto e_C$ is the universal
    cone valuation: For any simple valuation $\phi : \Cones_d \to G$, there is a
    homomorphism $h:\CG \to G$ such that $\phi = \phi_h$.
\end{theo}

By work of Gr\"omer~\cite{gromer} we can identify elements in $\Z\Cones_d / \mathcal{U}$ with
linear combinations of indicator functions $f = \sum_{i=1}^ka_i [C_i]$ where $C_1,\dots,C_k \in
\Cones_d$ and $a_1,\dots,a_k \in \Z$.

\begin{coro}\label{cor:almost}
    Let $f = \sum_i a_i[C_i]$ and $f' = \sum_i a'_i[C'_i]$. Then $f = f'$ in
    $\CG$ if and only if $f(p) = f'(p)$ for almost all $p \in \R^d$.
\end{coro}


\newcommand\Int[1]{\mathbb{T}_{#1}}
\newcommand\Ext[1]{\mathbb{O}_{#1}}

We will make extensive use of this correspondence.  In particular, we
can define the universal interior and exterior angle vectors. We will
describe them a little differently. Let $\CG[t]$ be the abelian group of
formal polynomials in $t$ with coefficients in $\CG$. For a polytope
$P$, we define
\[
    \Int{P}(t) \ \defeq \ \sum_{F} [\Tcone{F}{P}] \, t^{\dim F} \quad \text{ and
    } \Ext{P}(t) \ \defeq \ \sum_{F} [\Ocone{F}{P}] \, t^{\dim F}  \, ,
\]
where in both cases the sum is over all nonempty faces $F \subset P$. Thus, if $\CA$ is a cone
  angle, then $\aInt(P)$ is naturally identified with the coefficients of $\CA(\Int{P})$. Here is
  the first benefit.

\begin{prop}\label{prop:Ext0}
    Let $\CA$ be a cone angle and $P \subset \R^d$ a
    full-dimensional polytope. Then $\aExt_0(P)  =  1$.
\end{prop}
\begin{proof}
    Note that $\Ocone{v}{P} = \Ncone{v}{P}$ for any vertex $v \in P$.  Now,
    for a general $c \in \R^d$, the linear function $x \mapsto \inner{c,x}$
    will be maximized at a unique vertex of $P$. Corollary~\ref{cor:almost}
    implies
    \[
        \sum_{v} [\Ocone{v}{P}] \ = \ [\R^d]
    \]
    as elements in $\CG$. Applying $\alpha$ to both sides of the equation finishes the proof.
\end{proof}

To complete the first half of Theorem~\ref{thm:gram}, recall that the \Def{homogenization} of a
polytope $P \subset \R^d$ is the polyhedral cone
\[
    \hom(P) \ \defeq \ \cone( P \times \{1\}) \ = \ \{ (x,t) \in \R^d \times \R
    : \ t \ge 0, x \in t P \}
\]
Every face $\{0\} \neq F' \subseteq \hom(P)$ is of the form $\hom(F)$ for some nonempty face $F
  \subseteq P$. In particular, the definition of tangent cones extends to faces of $\hom(P)$.
  Moreover,
\[
    \Tcone{F}{P} \ \cong \ \Tcone{F'}{\hom(P)} \cap \{ (x,t) : t = 0 \} \, .
\]
Note that for $F' = \{0\}$, we have $\Tcone{F'}{\hom(P)} = \hom(P)$ and hence
\[
    \Tcone{F'}{\hom(P)} \cap \{ (x,t) : t = 0 \} \ = \ \{ (0,0) \}\,.
\]
On the other hand, if $F' = C$, then $\Tcone{C}{C} = \R^{d+1}$. A Brianchon-Gram relation for
  polyhedral cones was proved in~\cite{AS15}.

\begin{lemm}[{\cite[Lem.~4.1]{AS15}}]\label{lem:AS}
    Let $C \subseteq \R^{d+1}$ be a full-dimensional cone. Then
    as functions on $\R^{d+1}$
    \[
        \sum_{F'} (-1)^{\dim F'} [\Tcone{F'}{C}] \ = \
        (-1)^{d+1} [\mathrm{int}(-C)] \, ,
    \]
    where the sum is over all nonempty faces $F' \subseteq C$ and $\mathrm{int}(-C)$ denotes the
    interior of $-C$.
\end{lemm}

The following proposition proves the first half of Theorem~\ref{thm:gram}.

\begin{prop}\label{prop:Int0}
    Let $\CA$ be a cone angle on $\R^d$ and let $P \subset \R^d$ be a
    full-dimensional polytope. Then
    \[
        \aInt_0(P) - \aInt_1(P) + \aInt_2(P) - \cdots + (-1)^{d-1}
        \aInt_{d-1}(P) \ = \ (-1)^{d+1} \, .
    \]
\end{prop}

\begin{proof}
    Let $C = \hom(P) \subset \R^{d+1}$. This is a
    full-dimensional cone and Lemma~\ref{lem:AS} together with the restriction to $\R^d
    \times \{0\}$ and the preceding remarks yield the following relation on functions on
    $\R^d$
    \begin{equation}\label{eqn:gen_BR}
        [\{0\}] + \sum_{ F} (-1)^{\dim F+1}
        [\Tcone{F}{P}] + (-1)^{d+1} [\R^d] \ = \ (-1)^{d+1}[\emptyset] \, ,
    \end{equation}
    where the sum is over all nonempty faces $F \subseteq P$ with $F \neq P$. The above equation in
    $\CG$ reads
    \[
        (-1)^{d+1} [\R^d] \ = \
        \sum_{\emptyset \neq F \subset P} (-1)^{\dim F}[\Tcone{F}{P}]
        \ = \ \Int{P}(-1)
    \]
    and applying $\CA$ to both sides, yields the result.
\end{proof}

\section{Belt polytopes and angle vectors}\label{sec:zono}

A convex polytope $Z \subset \R^d$ is a \Def{zonotope} if there are $z_1,\dots,z_k \in \R^d
\setminus \{0\}$ and $t \in \R^d$ such that
\[
    t + Z \ = \ \sum_{i=1}^k [-z_i,z_i]
    \ = \ \{ \lambda_1 z_1 + \cdots + \lambda_k z_k : -1 \le
    \lambda_1,\dots,\lambda_k \le 1  \} \, .
\]
Zonotopes play an important role in geometric combinatorics~\cite[Ch.7]{ziegler} as well as
  convex geometry~\cite{GoodeyWeil}. Faces of zonotopes are zonotopes and hence all $2$-dimensional
  faces of a zonotope are centrally-symmetric polygons. In fact, this property characterizes
  zonotopes; see Bolker~\cite{Bolker}. A polytope $P \subset \R^d$ is a \Def{belt polytope} (or
  \Def{generalized zonotope}) if and only if every $2$-face $F \subset P$ has an even number of
  edges and opposite edges are parallel. Belt polytopes were studied by Baladze~\cite{Baladze} (see
  also~\cite{Bolker}) and are equivalently characterized by the fact that their normal fans are
  induced by hyperplane arrangements.

Let $\Arr$ be a central arrangement of hyperplanes, that is, the collection of some oriented linear
hyperplanes $H^0_i \defeq z_i^\perp$ for $i=1,\dots,k$. We write $H_i^+ = \{ x : \inner{z_i, x} >
0\}$ and $H_i^-$ accordingly. For a point $p \in \R^d$, let $\sigma_i = \sgn \inner{z_i,p}$ for
$i=1,\dots,k$. Then
\[
    H_\sigma \ \defeq \ H_1^{\sigma_1} \cap H_2^{\sigma_2} \cap \cdots \cap
    H_k^{\sigma_k}
\]
where $\sigma = (\sigma_1,\dots,\sigma_k) \in \{-,0,+\}^k$ is a relatively open cone containing
  $p$. This shows that the collection $\{ H_\sigma : \sigma \in \{-,0,+\}^k \}$ of relatively open
  cones partitions $\R^d$. The \Def{lattice of flats} of a hyperplane arrangement $\Arr$ is the
  collection of linear subspaces
\[
    \Lflats(\Arr) \ \defeq \ \{ H_{i_1}^0 \cap H_{i_2}^0 \cap \cdots  \cap
    H_{i_r}^0  : 1 \le i_1 < \cdots < i_r \le k, r \ge 0 \}
\]
partially ordered by \emph{reverse} inclusion. This is a graded lattice with minimal element
  $\R^d$ and maximal element $H_1^0 \cap \cdots \cap H_k^0$. For every relatively open cone
  $H_\sigma$, we have that
\[
    \affL(H_\sigma) \ = \ \bigcap_{i : \sigma_i = 0} H_i^0
\]
is an element of $\Lflats(\Arr)$ and we record the following consequence.

\begin{prop}\label{prop:arr_part}
    Let $\Arr$ be an arrangement of hyperplanes with $\Lflats =
    \Lflats(\Arr)$. Then any $L \in \Lflats$ is partitioned by the collection
    of relatively open cones $R$ of $\Arr$ with $\affL(R) \subseteq L$.
\end{prop}

Let $P \subset \R^d$ be a polytope of positive dimension and recall that for a non-empty face $F
\subseteq P$, we defined $\LL(F) = \affL(F)^\perp$. We can associate an arrangement $\Arr(P)$ with
hyperplanes $\LL(e)$ for every edge $e \subset P$. Every linear function $\ell(x) = \inner{c,x}$
yields a (possibly trivial) orientation on the edges of $P$ and thus determines a relatively open
cone of $\Arr(P)$. It can be shown that for every nonempty face $F \subset P$, the normal cone
$\Ncone{F}{P}$ is partitioned by some relatively open cones of $\Arr(P)$. In the language of
\cite[Sect.~7.1]{ziegler}, the fan induced by $\Arr(P)$ \emph{refines} the normal fan of $P$ and
the refinement is typically strict. It follows that $P$ is a belt polytope if and only if the
normal fan of $P$ coincides with the fan induced by $\Arr(P)$. The name `belt polytope' derives
from the following fact: two faces $F, F' $ of a belt polytope $P$ satisfy $\LL(F) = \LL(F')$ if
and only if $F$ and $F'$ are normally equivalent. The collection of faces $F$ with fixed $\LL(F)$
are said to be in the same belt.

Thus, if $P$ is a belt polytope, then $\Lflats(P) = \Lflats(\Arr(P))$ is a lattice graded by
dimension with minimum $\Lbot = \LL(v) = \R^d$ for every vertex $v$ and maximum $\Ltop = \LL(P)$.
See~\cite{Bolker} for details.

The \Def{Whitney numbers of the second kind} $W_i(\Lflats)$ of a graded poset $\Lflats$ count the
number of elements $a \in \Lflats$ of rank $\rk_\Lflats(a) = i$.

\begin{prop}\label{prop:zono_ext}
    Let $P$ be a belt polytope of dimension $d$ and let $\Lflats =
    \Lflats(P)$. Then for any cone angle $\CA$
    \[
        \aExt_i(P) \ = \ W_{i}(\Lflats)
    \]
    for all $i=0,\dots,d-1$.
\end{prop}

\begin{proof}
    Let $L \in \Lflats$. From Proposition~\ref{prop:arr_part} we infer that as
    elements of $\CG$
    \begin{equation}\label{eqn:ext_L}
        \sum_{F} [\Ocone{F}{P}]  \ = \ 
         \sum_{F} [L +\Ncone{F}{P}]  \ = \ 
        [\R^d] \, ,
    \end{equation}
    where the sum is over all faces $F \subseteq P$ with $\LL(F) = L$. For $i = 0,1,\dots,d-1$
    fixed it follows that
    \[
        \sum_{\substack{F \subset P\\ \dim F = i}} [\Ocone{F}{P}] \ = \
        \sum_{\substack{L \in \Lflats\\ \dim L = i}}  \
        \sum_{\substack{F \subset P\\ \LL(F) = L}}
        [\Ocone{F}{P}] \ = \
        \sum_{\substack{L \in \Lflats\\ \dim L = i}} [\R^d] \ = \
        W_{i}(\Lflats) [\R^d]
    \]
    and applying $\CA$ yields the claim.
\end{proof}

A configuration $z_1,\dots,z_n$ of $n \ge d$ vectors in $\R^d$ is \Def{generic} if any choice of
$d$ vectors are linearly independent. The proper faces of the associated zonotope $Z$ are
parallelepipeds. This implies that the poset $\Lflats(Z) \setminus \{\Ltop\}$ is isomorphic to the
collection of subsets of $[n]$ of cardinality at most $d-1$ ordered by inclusion and hence depends
only on $n$ and $d$.

\begin{coro}\label{cor:W-rel}
    For $d \ge 1$ and $\CA$ a cone angle
    \begin{align*}
      \aff \{ \aExt(P)  :  P \subset \R^d \text{ $d$-zonotope} \}
      \ &= \ \aff \{ \aExt(P)  :  P \subset \R^d \text{ $d$-polytope} \}\\
      \ &= \ \{ a \in \R^d : a_0 = 1 \} \, .
    \end{align*}
\end{coro}
\begin{proof}
    Proposition~\ref{prop:Ext0} implies that $\subseteq$ holds and
    thus we only need to exhibit $d$ zonotopes whose exterior angle vectors
    are linearly independent. By Proposition~\ref{prop:zono_ext}, it
    suffices to find $d$ zonotopes $Z_0,\dots,Z_{d-1} \subset \R^d$ such
    that the $d$-by-$d$ matrix ${A = (a_{ij})_{i,j=0,\dots,d-1}}$ with
    ${a_{ij} = W_i(Z_j)}$ has rank $d$. Let $Z_j$ be the zonotope obtained from a
    collection of $d+j$ generic vectors.  Then
    \[
        a_{ij} \ = \ \binom{d + j}{i} \quad \text{ for } i,j = 0,1,\dots,d-1
        \, .
    \]
    Row operations together with Pascal's identity then show that $A$ has determinant $1$, which
    proves the claim.
\end{proof}

The \Def{incidence algebra} $\Incidence(\poset)$ of a finite poset $(\poset, \preceq)$ is the
vector space of all functions ${h : \poset \times \poset \to \C}$, such that $h(a, c) = 0$ whenever
$a \not\preceq c$ and with multiplication
\[
    (g\ast h)(a, c) \  = \ \sum_{a \preceq b \preceq c} g(a, b)h(b, c)
\]
for $g, h \in \Incidence(\poset)$; see Stanley~\cite[Ch.~3]{EC1} for more on this. The \Def{zeta
  function} $\zeta_\poset \in \Incidence(\poset)$ is given by $\zeta_\poset(a,c) = 1$ if $a \preceq
  c$ and $=0$ otherwise. The zeta function is invertible in $\Incidence(\poset)$ with inverse given
  by the \Def{M\"obius function} $\mu_\poset = \zeta_{\poset}^{-1}$. More precisely, the M\"obius
  function satisfies $\mu_\poset(a,a) = 1$ and
\[
    \mu_\poset(a,c) \ = \ -\sum_{a \prec b \preceq c} \mu_\poset(b,c)
\]
for $a \prec c$. For a graded poset $\poset$ of rank $d$, the \Def{characteristic polynomial}
  ${\chi_\poset(t) \in \Z[t]}$ is defined by
\[
    \chi_\poset(t) \ = \ \sum_{a \in \poset} \mu_{\poset}(\Lbot,a) \, t^{d -
            \rk(a)}  \ = \ w_0(\poset) t^d + w_1(\poset) t^{d-1} + \cdots +
    w_d(\poset) \, .
\]
The numbers $w_i(\poset)$, called the \Def{Whitney numbers of the first kind}, are explicitly
  given by
\[
    w_i(\poset) \ = \ \sum_{a \,:\, \rk(a) = i} \mu_\poset(\Lbot,a) \, .
\]
\newcommand\CoChar{\psi}
In particular, $w_0(\poset) = 1$ and $w_d(\poset) = \mu_\poset(\Lbot,\Ltop)$.
The characteristic polynomial $\chi_\Lflats(t)$ where $\Lflats =
\Lflats(\Arr)$ is the lattice of flats of a hyperplane arrangement $\Arr$
captures a number of important properties. For example, Zaslavsky's celebrated
result~\cite{Zaslavsky} states that $|\chi_\Lflats(-1)|$ is the number of
regions of $\Arr$; see also~\cite[Ch.~3.6]{crt}. Here, however, we will be
interested in the characteristic polynomial of the opposite poset
$\Lflats^\op$.

\begin{lemm}\label{lem:key}
    Let $P$ be a $d$-dimensional belt polytope with lattice of flats $\Lflats
    = \Lflats(P)$ and let $\CA$ be a cone angle.  For any fixed $L \in
    \Lflats$
    \[
        \sum_{F} \aInt(F,P) \ = \
        (-1)^{d-\dim L}\mu_{\Lflats}(L,\Ltop) \ = \
        (-1)^{d-\dim L}\mu_{\Lflats^\op}(\Lbot,L) \, ,
    \]
    where the sum is over all faces $F \subseteq P$ with $\LL(F) = L$.
\end{lemm}

The proof makes use of the fact that tangent and normal cones are related by polarity.

\begin{prop}\label{prop:NT_polar}
    Let $P \subset \R^d$ be a full-dimensional polytope and $v \in P$ a
    vertex. Then
    \[
        (\Ncone{v}{P})^\polar \ = \ \{ u \in \R^d : \inner{u,x} \le 0 \text{
            for all } x \in \Ncone{v}{P} \} \ = \ \Tcone{v}{P} \, .
    \]
\end{prop}
\begin{proof}
    \newcommand\cc{\mathbf{c}}
    Observe that $\cc \in \Ncone{v}{P}$ if and only if $\inner{\cc,v} \ge
    \inner{\cc,x}$ for all $x \in P$. That is, if and only if $\inner{\cc,x -
        v} \le 0$ for all $x \in P$ and from $\Tcone{v}{P} = \cone(-v + P)$
    we deduce that $\Tcone{v}{P}^\polar = \Ncone{v}{P}$.
\end{proof}

\begin{proof}[Proof of Lemma~\ref{lem:key}]
    We again prove the following more general statement over $\CG$
    \begin{equation}\label{eqn:GZ}
        \sum_{F} [\Tcone{F}{P}] \ = \
        (-1)^{d-\dim L} \mu_{\Lflats(P)}(L,\Ltop)  \, [\R^d] \, ,
    \end{equation}
    where the sum is over all faces $F \subseteq P$ with $\LL(F) = L$.

    Let us assume that $L = \Lbot = \{0\}$. Proposition~\ref{prop:NT_polar} states that
    $\Tcone{\v}{P}$ is precisely the polar cone $\Ncone{\v}{P}^\polar$. That is, if $\w \in
    \interior(\Tcone{\v}{P})$, then the hyperplane $\w^\perp$ does not meet
    $\interior(\Ncone{\v}{P})$. Note that since $P$ is a belt polytope, the cones $\Ncone{\v}{P}$
    are the regions of $\Arr = \Arr(P)$.

    Hence, for a generic $\w$, the left-hand side of~\eqref{eqn:GZ} is the number of regions of
    $\Arr$ that are not intersected by $\w^\perp$. By a classical result of Greene and
    Zaslavsky~\cite[Thm.~3.1]{GZ}, this number is independent of $\w$ and is exactly $(-1)^{d-\dim
    L}\mu_{\Lflats(P)}(\Lbot,\Ltop)$.

    For $L \neq \Lbot$, let $\pi_L : \R^d \to L^\perp$ be the orthogonal projection along $L$. Then
    $\pi_L(P)$ is a belt polytope and $\Lflats(\pi_L(P))$ is isomorphic to the interval $[L,\Ltop]
    \subseteq \Lflats(P)$.
\end{proof}

The following shows that the interior angle vectors of zonotopes are determined by the Whitney
numbers of the first kind. Together with Proposition~\ref{prop:zono_ext}, this proves
Theorem~\ref{thm:whitney}.

\begin{proof}[Proof of Theorem~\ref{thm:whitney}]
    Let $P$ be a belt polytope with lattice of flats $\Lflats = \Lflats(P)$.
    With the help of Lemma~\ref{lem:key}, we deduce for $L \in \Lflats$
    with $\dim L = i$
    \begin{align*}
      \aInt_i(P)
      & = \ \sum_{\dim F = i} \aInt(F,P)
      \ = \ \sum_{\dim L = i} \sum_{\LL(F) = L}\aInt(F,P)
      \ = \ \sum_{\dim L = i} (-1)^{d-i}\mu_{\Lflats^\op}(\Lbot,L)\\
      & = \ (-1)^{d-i}w_{d-i}(\Lflats^\op). \qedhere
    \end{align*}
\end{proof}

In~\cite{NPS}, Novik, Postnikov, and Sturmfels introduced the \Def{cocharacteristic polynomial} of
the lattice of flats: For a zonotope $Z$ of dimension $d$ and lattice of flats $\Lflats =
\Lflats(Z)$, its cocharacteristic polynomial is
\[
    \CoChar_\Lflats(t) \ = \ \sum_{L \in \Lflats} |\mu_\Lflats(L,\Ltop)| \,
    t^{d - \dim L} \ = \ \sum_{i=0}^{d}  |w_{d-i}(\Lflats^\op)| t^{d-i} \ = \
    (-t)^d \chi_{\Lflats^\op}(-\tfrac{1}{t}) \, .
\]
In~\cite{NPS} the coefficients of the cocharacteristic polynomial encoded invariants of ideals
  associated to matroids. Here, cocharacteristic polynomials give us an elegant mean to prove the
  following theorem, which proves the uniqueness of~\eqref{eqn:gram} in Theorem~\ref{thm:gram}.

\begin{theo}\label{thm:w-rel}
    For $d \ge 1$, let $\CA : \Cones_d \to \R$ be a cone angle. Then
    \[
        \aff \{ \aInt(Z)  :  Z \subset \R^d \text{ $d$-zonotope} \} =
        \Big\{ (a_0,\dots,a_{d-1}) \in \R^d : \sum_{i = 0}^{d-1} (-1)^i a_i = (-1)^{d+1} \Big\}.
    \]
\end{theo}
\begin{proof}

    Using Theorem~\ref{thm:whitney}, it suffices to produce $d$ zonotopes $Z_0,\dots,Z_{d-1}$ whose
    cocharacteristic polynomials are linearly independent.

    For $j \ge 0$, let $Z_j$ be the $d$-dimensional zonotope of $d+j$ generic vectors and let
    $\CoChar_{d,j}(t)$ be its associated cocharacteristic polynomial. From~\cite[Prop.~4.2]{NPS} we
    deduce that these polynomials satisfy the recurrence
    \[
        \CoChar_{d,j}(t) \ = \ \CoChar_{d-1,j}(t)  + \binom{d-1+j}{j} t
        (t+1)^{d-1}
    \]
    for $d \ge 1$ and $\CoChar_{0,j}(t) = 1$. We claim that the cocharacteristic polynomials
    $\CoChar_{d,j}(t)$ for ${0 \le j \le d-1}$ are \renewcommand\l{\lambda} linearly independent.
    Indeed, the recursion and the fact that $\deg \CoChar_{d,j}(t) = d$ shows that $\sum_j \l_j
    \CoChar_{d,j} = 0$ for $\l_0,\dots,\l_{d-1} \in \R$ if and only if
    \[
        \sum_{j=0}^{d-1} \binom{d-1+j}{j} \l_j \ = \ 0 \quad \text{ and } \quad
        \sum_{j=0}^{d-1} \CoChar_{d-1,j}(t) \l_j \ = \ 0 \, .
    \]
    Iterating this idea, it follows that $\l = (\l_0,\dots,\l_d)$ is in the kernel of the
    $d$-by-$d$ matrix $A$ with entries $\binom{i+j}{j}$ for $i,j = 0,\dots,d-1$. Again appealing to
    Pascal's identity, it is easy to see that $\det A = 1$, which completes the proof.
\end{proof}

\section{Connecting angles with M\"obius inversion}\label{sec:incalg}

In this section we take an algebraic approach to the occurrence of the Whitney numbers of the
lattice of flats of a belt polytope in the previous sections. The \Def{face lattice} of a polytope
$P$ is the collection $\Lfaces(P)$ of faces of $P$ ordered by inclusion. For a given belt polytope
$P$, we define a certain subalgebra of $\Incidence(\Lfaces(P))$. As it will turn out the map $F
\mapsto \LL(F)$ yields a pair of transformations and Theorem~\ref{thm:whitney} follows from the
fact that the two transformations are adjoint. In particular, we derive a generalization of a
result of Klivans and Swartz~\cite{KlivansSwartz} regarding spherical intrinsic volumes and Whitney
numbers.

\newcommand{\posetq}{\mathcal{Q}}

Let $\poset, \posetq$ be two posets. A surjective and order preserving map
$\phi : \poset \to \posetq$ induces a linear transformation $\phi_\ast :
\Incidence(\poset) \to \Incidence(\posetq)$ by
\[
    \phi_\ast h(q,q') \ \defeq \ \ \frac{1}{|\phi^{-1}(q')|}
    \sum_{ \substack{p \in \phi^{-1}(q)\\ p' \in \phi^{-1}(q')}} h(p,p')
\]
called the \Def{pushforward} of $h$. Let $\Incidence_\phi(\poset) \subseteq \Incidence(\poset)$
  be the vector subspace of all elements $h \in \Incidence(\poset)$ such that for all $q,q' \in
  \posetq$
\begin{equation}\label{eqn:phi}
   \sum_{p \in \phi^{-1}(q)} h(p, p_1') \ = \ \sum_{p \in \phi^{-1}(q)} h(p,
   p_2')
   \qquad \text{ for all } \ p'_1,p'_2  \in \phi^{-1}(q') \, .
\end{equation}
The neutral element $\delta \in \Incidence(\poset)$ is defined by $\delta(x,y) = 1$ if $x=y$, and
$=0$ otherwise. Clearly, $\delta \in \Incidence_\phi(\poset)$ and thus $\Incidence_\phi(\poset)
\neq \emptyset$. For an element $h \in \Incidence_\phi(\poset)$, the pushforward simplifies to
\[
    \phi_\ast h(q,q') \ = \ \sum_{p \in \phi^{-1}(q)} h(p,p')
\]
for any $p' \in \phi^{-1}(q')$.

\begin{prop}\label{prop:Rf_is_alg}
    $\Incidence_\phi(\poset)$ is a subalgebra of $\Incidence(\poset)$ and
    $\phi_\ast : \Incidence_\phi(\poset) \to \Incidence(\posetq) $ is an
    algebra map.
\end{prop}
\begin{proof}
    Let $g, h \in \Incidence_\phi(\poset)$. For $q, q' \in \posetq$
    and $p' \in \phi^{-1}(q')$ arbitrary we compute
    \begin{align*}
            (\phi_\ast g \ast \phi_\ast h)(q, q') 
            \ &= \
              \sum_{s \in \posetq} \phi_\ast g(q, s) \cdot \phi_\ast h(s, q') 
            \ = \
            \sum_{p \in \phi^{-1}(q)} \sum_{s \in \posetq} \sum_{r \in \phi^{-1}(s)} g(p,
            r) \cdot h(r, p')\\
            \ &= \
            \sum_{p \in \phi^{-1}(q)} \sum_{r \in \poset} g(p, r) \cdot h(r, p')
            \ = \ 
            \sum_{p \in \phi^{-1}(q)} (g \ast h)(p, p') \, .
    \end{align*}
    Since the left-hand side does not depend on the choice of $p'$, we see that $g \ast h \in
    \Incidence_\phi(\poset)$ and therefore $\phi_\ast g \ast \phi_\ast h = \phi_\ast (g \ast h)$.
    Since $\delta \in \Incidence_\phi(\poset)$, this shows that $\Incidence_\phi(\poset)$ is a
    subalgebra.
\end{proof}

For a graded poset $\poset$ of rank $d$, we can define a binary operation
\[
    \ast_k : \Incidence(\poset) \times \Incidence(\poset) \to \Incidence(\poset)
\]
for $k = 0,\dots,d$ by
\begin{equation}\label{eqn:k-prod}
    (g \ast_k h)(a,c) \ \defeq \ \sum_{b \,:\, \rk(b)=k} g(a,b) \, h(b,c) \, .
\end{equation}
By definition $g \ast h = \sum_k g \ast_k h$. It is noteworthy that $\ast_k$ and $\ast$ are
associative operations, i.e., for $0 \leq k \leq l \leq d$ and $g,h,m \in \Incidence(\poset)$
\[
    g \ast (h \ast_k m) = (g \ast h) \ast_k m \, , \qquad
    g \ast_k (h \ast m) = (g \ast_k h) \ast m \, , \qquad
    g \ast_k (h \ast_l m) = (g \ast_k h) \ast_l m \, .
\]

The proof of Proposition~\ref{prop:Rf_is_alg} carries over verbatim to prove the following
corollary.

\begin{coro}\label{cor:pushfor_k}
    Let $\poset$ and $\posetq$ be ranked posets. If $\phi : \poset \to
    \posetq$ is a surjective order preserving map that preserves rank, then
    \begin{equation}\label{eqn:pushfor_k}
        \phi_\ast( g \ast_k h) \ = \ \phi_\ast g  \ast_k \phi_\ast h \, .
    \end{equation}
\end{coro}

Let $C \subset \R^d$ be a polyhedral cone and let $\Lfaces_+(C)$ the collection of nonempty faces
of $C$ partially ordered by inclusion. For a given cone angle $\CA$, we note that the interior and
exterior angles
\begin{align*}
  \aInt(F,G) \ &= \ \CA(\Tcone{F}{G})  \ = \ \CA(\Tcone{F}{G} + \affL(G)^\perp)\\
  \aExt(F,G) \ &= \ \CA(\Ocone{F}{G}) \ = \ \CA(\Ncone{F}{G} + \affL(F) )\\
\end{align*}
for faces $F \subseteq G \subseteq C$ are naturally elements of the incidence algebra
${\Incidence(C) \! \defeq \!\Incidence(\Lfaces_+(C))}$. Furthermore, let us define:
\begin{align*}
  \aInt'(F,G) \ \defeq \ (-1)^{\dim G - \dim F} \aInt(F,G)\,.
\end{align*}

We call two cone angles $\CA, \CAb$ \Defn{complementary} if for all polyhedral cones $C \subseteq
\R^d$
\begin{equation}\label{eqn:compl}
    \aInt' \ast \bExt \ = \ \delta_C \, ,
\end{equation}
where $\delta_C$ is the neutral element of $\Incidence(C)$. As a notable example, the standard cone
angle $\SCA$ is self-complementary, i.e. $\widehat{\SCA}' \ast \widecheck{\SCA} = \delta_C$ for all
polyhedral cones $C \subseteq \R^d$; see~\cite{McM-angle}. Complementary angles were studied by
McMullen~\cite{McM-polytopealgebra} under the name of \emph{inverse} angles.
In~\cite{McM-polytopealgebra}, an \Defn{angle functional} is a collection of normalized and simple
valuations $\CA_L$ for cones in linear subspaces $L \subseteq \R^d$. The angle of a cone $C
\subseteq \R^d$ is then $\CA_L(C)$ where $L = \affL(C)$. In this framework, we define the cone
angle $\CA_L$ on a linear subspace $L \subset \R^d$ by $\CA_L(C) \defeq \CA(C + L^\perp)$ for any
cone $C \subseteq L$. The next lemma is an adaptation of~\cite[Lemma~46]{McM-polytopealgebra}.

\begin{lemm}\label{lem:comp_angle}
    For every cone angle $\CA$ there is a complementary cone angle $\CAb$.
\end{lemm}
\begin{proof}
    Lemma~46 in~\cite{McM-polytopealgebra} guarantees the existence of
    a complementary angle functional $\beta_L$ for the angle functional $\alpha_L$
    as defined above, such that~\eqref{eqn:compl} is satisfied.  It
    can be shown that $\beta_L$ is of the form
    $\CAb_L(C) = \CAb(C + L^\perp)$ for some cone angle $\CAb$.
\end{proof}

\newcommand\newMin{\bot}

Let $P \subset \R^d$ be a belt polytope and let $\Lfaces = \Lfaces(P)$ be the
collection of faces of $P$. As before, we can interpret $\aInt(F,G)$ and
$\aExt(F,G)$ as elements in $\Incidence(\Lfaces)$, by extending
$\aInt(\emptyset,G) = 1$ if $\dim G \le 0$ and $=0$ otherwise and
$\aExt(\emptyset,G) = 1$ for all $G$. In particular, $\aInt' \ast \bExt =
\delta_\Lfaces$.

Let $\Lflats_0$ be the set $\Lflats(P) \cup \{ \newMin \}$ partially ordered by inclusion. Recall
that for a non-empty face $F \subseteq P$, $\LL(F) = \affL(F)^\perp$, where $\affL(F)$ is the
linear subspace parallel to $F$. Setting $\LL(\emptyset) \defeq \newMin$, the map $\LL : \Lfaces(P)
\to \Lflats_0(P)$ given by $F \mapsto \LL(F)$ is a surjective order and rank preserving map.

\begin{theo}\label{thm:push}
    Let $P$ be a belt polytope and $\Lfaces = \Lfaces(P)$.
    For every cone angle $\CA$, we have $\aInt, \aExt \in
    \Incidence_{\LL}(\Lfaces)$ and
    \[
        \LL_\ast \aExt \ = \  \zeta_{\Lflats_0}
        \quad \text{ and } \quad
        \LL_\ast \aInt' \ = \ \mu_{\Lflats_0} \, .
    \]
\end{theo}
\begin{proof}
    Two faces $G$ and $G'$ of $P$ are normally equivalent if $\LL(G) = \LL(G')$.
    Thus equation~\eqref{eqn:phi} is satisfied and $\aExt$ and $\aInt$ are
    elements of $\Incidence_{\LL}(\Lfaces)$.  Let $G \subseteq P$ be a face and
    $U \in \Lflats$ with $U \supseteq \LL(G)$. Then from~\eqref{eqn:ext_L} in
    the proof of Proposition~\ref{prop:zono_ext} and $\aExt(\emptyset, G) = 1$
    we infer that
    \[
        \sum_{\substack{F \in \Lfaces(P), \\ \LL(F) = U}} \aExt(F,G) = 1
    \]
    and hence $\big(\LL_\ast \aExt \big) (U,U') = 1 = \zeta(U, U')$ for all $U,U' \in \Lflats$ with
    $U \supseteq U'$.

    By Lemma~\ref{lem:comp_angle}, there is a cone angle $\beta$ complementary to $\alpha$. Using
    the fact that $\LL_*$ is an algebra map, we deduce
    \[
        \delta_{\Lflats_0} \ = \ \LL_\ast(\delta_{\Lfaces}) \ = \ \LL_\ast(\aInt' \ast
        \bExt) \ = \ \LL_\ast(\aInt') \ast \LL_\ast(\bExt) \;.
    \]
    Replacing $\aExt$ by $\bExt$ above yields $\LL_\ast(\bExt) = \zeta_{\Lflats}$ and thus
    $\LL_\ast(\aInt) = \zeta_{\Lflats}^{-1} = \mu_{\Lflats}$.
\end{proof}

Recall that $\SCA(C) = \frac{\vol(C \cap B_d)}{\vol(B_d)}$ is the standard cone angle. For a
polytope $P \subset \R^d$, the $k$-th \Def{spherical intrinsic volume} is defined as
\begin{equation}\label{eqn:sp_int_vol}
    \overline{\SCA}_k(P) \ \defeq \ 
    \sum_{v}
    \sum_{v \in F}
    \widehat{\SCA}(v,F)
    \widecheck{\SCA}(F,P) \, ,
\end{equation}
\newcommand\aSph{\overline{\CA}} where the sum is over all vertices $v \in P$ and $k$-faces $F \subset P$. For a
given cone angle $\CA$, we denote by $\aSph_k(P)$ the generalization
of~\eqref{eqn:sp_int_vol} to $\CA$.

The machinery developed in this section yields algebraic proofs of Theorem~\ref{thm:whitney}.

\begin{coro} \label{cor:alg-KS}
    Let $\CA$ be a cone angle and $P$ a $d$-dimensional belt polytope.  For $k
    = 0,\dots,d-1$ the following hold:
    \begin{enumerate}[\rm (i)]
        \item $\aExt_k(P)  =  W_{k}(\Lflats(P))$;
        \item $\aInt_k(P)  =  |w_{d-k}(\Lflats(P)^\op)|$;
        \item $\aSph_k(P)  = |w_{k}(\Lflats(P))|$.
    \end{enumerate}
\end{coro}

Parts (ii) and (iii) were shown by Klivans and Swartz in~\cite{KlivansSwartz} for the standard cone
angle; see also~\cite{AL, Schneider17}. For the proof, we need the following technical result.

\begin{lemm}\label{lem:sum_of_rank}
    Let $\phi : \poset \to \posetq$ be a surjective, order and rank preserving
    map between posets with minimal and maximal elements.  If  $f \in
    \Incidence_\phi(\poset)$, then
    \[
        \big(\zeta_\poset \ast_k f\big)(\Lbot_\poset, \Ltop_\poset) \ = \
        \big(\zeta_\posetq \ast_k (\phi_* f)\big)(\Lbot_\posetq, \Ltop_\posetq)\,.
    \]
\end{lemm}
\begin{proof}
    Writing out the definition of $\zeta_\poset \ast_k f$ we obtain
    \begin{align*}
     \big(\zeta_\poset \ast_k f\big)(\Lbot_\poset, \Ltop_\poset)
       & \ = \ \sum_{\substack{p \in \poset \\ \rk(p) = k}} f(p, \Ltop_\poset)
       \ = \ \sum_{\substack{q \in \posetq \\ \rk(q) = k}} \sum_{p \in \phi^{-1}(q)} f(p, \Ltop_\poset)\\
     & \ = \ \sum_{\substack{q \in \posetq \\ \rk(q) = k}} \big(\phi_* f\big)(q, \Ltop_\posetq)
       \ = \ \big(\zeta_\posetq \ast_k \phi_* f\big)(\Lbot_\posetq, \Ltop_\posetq)\,.\qedhere
   \end{align*}
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:alg-KS}]
    (i) immediately follows from
    Theorem~\ref{thm:push} and
    Lemma~\ref{lem:sum_of_rank} for $k = i$ and $f = \aExt$. Relation
    (ii) follows in the same fashion with $f = \aInt$, but note that we obtain the
    co-Whitney numbers of the first kind. For (iii), we invoke the
    same lemma for $k = 0$ and $f = \aInt \ast_i \aExt$.
\end{proof}

This algebraic perspective on angles is very helpful and will facilitate proofs and computations in
the next sections.

\section{Flag-angle vectors}\label{sec:flag}

In this and the next section we prove Theorems~\ref{thm:flag_rels} and~\ref{thm:flag_whitney}. Our
strategy of proof is as follows. First, we will show that the interior/exterior flag-angle vectors
satisfy the relations stated in Theorem~\ref{thm:flag_rels}. This is done in
Propositions~\ref{prop:flag_ext_rels} and~\ref{prop:flag_int_rels}. The algebraic machinery
developed in Section~\ref{sec:incalg} enables us to prove Theorem~\ref{thm:flag_whitney}. To
complete the proof of Theorem~\ref{thm:flag_rels}, we use this combinatorial interpretation of
flag-angle vectors for belt polytopes. It suffices to show that there are no linear relations on
flag-Whitney numbers of lattices of flats. For the flag-Whitney numbers of the second kind, this is
done in Section~\ref{sec:flag-whitney} and, by establishing an algebraic connection
(Theorem~\ref{thm:unipot}) between them, this also addresses the case of flag-Whitney numbers of
the first kind.

The following is the analogue of Proposition~\ref{prop:Ext0}.
\begin{prop}\label{prop:flag_ext_rels}
    Let $P$ be a $d$-dimensional polytope and $S \subseteq [d-1]$. Then
    \[
        \aExt_{S}(P) \ = \ \aExt_{S \cup \{0\}}(P) \, .
    \]
\end{prop}
\begin{proof}
    Let $S = \{s_1,\dots,s_k\}$ and set $s_0 \defeq 0$.  Unravelling the
    definition of exterior flag-angle vectors (see~\eqref{eqn:flag}), we compute
    \begin{align*}
        \aExt_{S \cup \{0\}}(P) \ &= \ 
     \sum_{F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_k}
    \aExt(F_0,F_1) \,  \aExt(F_1,F_2) \cdots \aExt(F_k,P) \\
    \ &= \
     \sum_{F_1 \subset F_2 \subset \cdots \subset F_k}
     \,  \aExt(F_1,F_2) \cdots \aExt(F_k,P) 
    \sum_{F_0 \subset F_1} \aExt(F_0,F_1)\\
    \ &= \
     \sum_{F_1 \subset F_2 \subset \cdots \subset F_k}
     \,  \aExt(F_1,F_2) \cdots \aExt(F_k,P) \\
     &= \ \aExt_{S}(P) \, ,
    \end{align*}
    where the sums are over faces $F_i$ with $\dim F_i = s_i$ for $i=0,\dots,k$ and where the third
    equality follows from Proposition~\ref{prop:Ext0}.
\end{proof}

As for the linear relations on \emph{interior} flag-angle vectors, we take a more algebraic
approach. Let $P$ be a $d$-dimensional polytope with face lattice $\Lfaces = \Lfaces(P)$ and ${S =
\{s_1 < s_2 < \cdots < s_k\}}$ with $S \subseteq [0, d-1]$. Using~\eqref{eqn:k-prod} together with
the fact that $\rk_\Lfaces(F) = \dim F + 1$, we can give the following expression for the $S$-entry
of the interior flag-angle vector
\[
    \aInt_{S}(P) \ = \
    (\zeta_{\Lfaces}  \ast_{s_1+1}
    \aInt \ast_{s_2+1}
    \cdots \ast_{s_k+1} \aInt)(\emptyset,P)   \, ,
\]
where the operation $\ast_k$ was introduced in~\eqref{eqn:k-prod}.

\begin{prop}\label{prop:flag_int_rels}
    Let $P$ be a $d$-polytope. For $S = \{0 \le s_1 < s_2 <
    \cdots < s_k \le d-1 \}$ set $t \defeq \min(S \cup \{d\})$. Then
    \[
        \sum_{i=0}^{t-1} (-1)^{i} \aInt_{S \cup \{i\}}(P) \ = \
        (-1)^{t+1} \aInt_S(P) \, .
    \]
\end{prop}

\begin{proof}
    Recall that the M\"obius function $\mu_\Lfaces = \zeta_{\Lfaces}^{-1}$ is
    given by
    \[
        \mu_{\Lfaces}(F,G) = (-1)^{\dim G - \dim F}
    \]
    for faces $F \subseteq G \subset P$. For a fixed face $G$, Proposition~\ref{prop:Int0} yields
    \[
        (\mu_\Lfaces \ast \aInt)(\emptyset,G) \ = \ - \sum_{F} (-1)^{\dim F}
        \aInt(F,G) \ = \ -\kern-3pt \sum_{i=0}^{\dim G-1} (-1)^i \aInt_i(G) +
        (-1)^{\dim G+1} \ = \ 0 \, .
    \]
    The result now follows by evaluating
    \[
        \mu_{\Lfaces} \ast \aInt  \ast_{s_1+1} \aInt \ast_{s_2+1} \cdots
        \ast_{s_k+1} \aInt
    \]
    at $(\emptyset,P)$.
\end{proof}

Let $\poset$ be a graded poset of rank $d+1$ and let $S = \{s_1 < s_2 < \dots < s_k\} \subseteq
[d]$. The \Def{flag-Whitney numbers of the second kind} as defined in the introduction are given by
\[
    W_S(\poset) \ = \ ( \zeta_\poset \ast_{s_1} \zeta_\poset \ast_{s_2} \cdots
    \ast_{s_k} \zeta_\poset) (\Lbot,\Ltop) \, .
\]
Similarly, we define the \Defn{flag-Whitney numbers of the first kind} by
\[
    w_S(\poset) \ \defeq \
    ( \mu_\poset \ast_{s_1} \mu_\poset \ast_{s_2} \cdots
    \ast_{s_k} \zeta_\poset) (\Lbot,\Ltop) \ = \
    \sum \mu(\Lbot, c_1) \mu(c_1, c_2) \cdots \mu(c_{k-1}, c_{k})\; ,
\]
where the sum is over all chains $\Lbot \prec c_1 \prec c_2 \prec \cdots \prec c_k$ with $\rk c_i
  = s_i$ for $i=1,\dots,k$. Now the same reasoning as in the proof of Corollary~\ref{cor:alg-KS}
  yields Theorem~\ref{thm:flag_whitney}:

\begin{proof}[Proof of Theorem~\ref{thm:flag_whitney}]
    Let $\Lfaces = \Lfaces(P)$ be the face lattice of $P$ and let $\Lflats_0$ be
    the poset $\Lflats(P)$ with a new minimal element $\Lbot_{\Lflats_0}$
    adjoined. The maximal element of $\Lflats_0$ is $\Ltop_{\Lflats_0} =
    \LL(P)$.  We also set $t_i \defeq s_i + 1$ and $f \defeq \aExt \ast_{t_2}
    \dots \ast_{t_k} \aExt$. Note that $f \in \Incidence_\LL(\Lfaces)$ and with
    Lemma~\ref{lem:sum_of_rank} we compute
    \begin{align*}
     \aExt_S(P)
     &\ = \ \big(\zeta_\Lfaces \ast_{t_1} \aExt \ast_{t_2} \dots \ast_{t_k} \aExt\big)(\emptyset, P)\\
     &\ = \  \big(\zeta_\Lfaces \ast_{t_1} f\big)(\Lbot_{\Lfaces}, \Ltop_{\Lfaces})
      \ = \ \big(\zeta_\Lfaces \ast_{t_1} (\LL_\ast f)\big)(\Lbot_{\Lflats_0}, \Ltop_{\Lflats_0})\\
     &\stackrel{\eqref{eqn:pushfor_k}}{\ = \ }  \big(\zeta_{\Lflats_0} \ast_{t_1} \zeta_{\Lflats_0} \ast_{t_2} \dots \ast_{t_k} \zeta_{\Lflats_0}\big)(\Lbot_{\Lflats_0}, \Ltop_{\Lflats_0})\\
     &\ = \ \big(\zeta_{\Lflats} \ast_{s_1} \zeta_{\Lflats} \ast_{s_2} \dots \ast_{s_k} \zeta_{\Lflats}\big) (\Lbot_{\Lflats}, \Ltop_{\Lflats})\\
     &\ = \ W_{S}(\Lflats) \ = \ W_{d-S}(\Lflats^\op) \,.
   \end{align*}
    Similarly for the second statement, for $g \defeq \aInt \ast_{t_2} \dots \ast_{t_k} \aInt \in
    \Incidence_\LL(\Lfaces)$ we obtain:
    \begin{align*}
     \aInt_S(P)
     &\ = \ \big(\zeta_\Lfaces \ast_{t_1} \aInt \ast_{t_2} \dots \ast_{t_k} \aInt\big)(\emptyset, P)\\
     &\ = \ \big(\zeta_\Lfaces \ast_{t_1} g\big)(\Lbot_{\Lfaces}, \Ltop_{\Lfaces})
     \  = \ \big(\zeta_\Lfaces \ast_{t_1} (\LL_\ast g)\big)(\Lbot_{\Lflats_0},
       \Ltop_{\Lflats_0})\\
     &\stackrel{\eqref{eqn:pushfor_k}}{\ = \ } (-1)^{d + 1 - t_1} \cdot \big(\zeta_{\Lflats_0} \ast_{t_1} \mu_{\Lflats_0} \ast_{t_2} \dots \ast_{t_k} \mu_{\Lflats_0}\big)(\Lbot_{\Lflats_0}, \Ltop_{\Lflats_0})\\
     &\ = \ (-1)^{d - s_1} \cdot \big(\zeta_{\Lflats} \ast_{s_1} \mu_{\Lflats} \ast_{s_2} \dots \ast_{s_k} \mu_{\Lflats}\big) (\Lbot_{\Lflats}, \Ltop_{\Lflats})\\
     &\ = \ (-1)^{d-s_1} \cdot w_{d-S}(\Lflats(P)^\op)\,.\qedhere
    \end{align*}
\end{proof}

\newcommand\Poly{\mathcal{P}}
\newcommand\dbrackets[1]{[\![#1]\!]}
In order to complete the proof of Theorem~\ref{thm:flag_rels}, we observe that
the flag-Whitney numbers of the second kind determine the flag-Whitney numbers
of the first kind. We show this in more generality. Let $\poset$ be a finite
poset with $\Lbot$ and $\Ltop$ and let ${R \defeq \C\dbrackets{z_a : a \in \poset}}$
be the ring of formal power series with variables indexed by elements
of $\poset$.  For a unipotent $g \in \Incidence(\poset)$, i.e., $g(a,a) = 1$
for all $a \in \poset$, we define
\[
    F_g(\z) \ \defeq \ \sum
    g(\Lbot, c_1) \, z_{c_1} \,
    g(c_1, c_2) \, z_{c_2} \,
    \cdots
    \, z_{c_{k-1}} \, g(c_{k-1}, c_k) \, z_{c_k} \, ,
\]
where the sum is over all multichains $\Lbot \prec c_1 \preceq c_2 \preceq \cdots \preceq c_k
  \prec \Ltop$. Since every multichain comes from a unique chain, we can rewrite $F_g(\z)$ to
\[
    F_g(\z) \ = \
    \sum_{\Lbot \prec b_1 \prec b_2 \prec \cdots \prec b_k \prec \Ltop}
    g(\Lbot, b_1) \frac{z_{b_1}}{1 - z_{b_1}}
    g(b_1, b_2) \frac{z_{b_2}}{1 - z_{b_2}}
    \cdots
    g(b_{k-1}, b_k)
    \frac{z_{b_k}}{1 - z_{b_k}}    \, .
\]

If $\poset$ is a graded poset of rank $d+1$, then for $g = \zeta$, we get
\[
    G_\poset(\q) \ \defeq \ F_\zeta(z_a = q_{\rk(a)} : a \in \poset) \ = \
    \sum_{S \subseteq [d]} W_S(\poset) \prod_{i \in S} \frac{q_i}{1-q_i} \ \in
    \ \C\dbrackets{q_1,\dots,q_d} \, .
\]
Since the elements $\frac{q_i}{1-q_i}$ for $i=1,\dots,d$ are algebraically independent over
  $\C\dbrackets{q_1,\dots,q_d}$, $G_\poset(\q)$ encodes the flag-vector of $\poset$. The relation
  to the flag-Whitney numbers of the second kind follows from the next theorem.

\begin{theo}\label{thm:unipot}
    Let $g \in \Incidence(\poset)$ be  unipotent.  Then
    \[
        F_g(\tfrac{1}{\z}) \ = \ F_{g^{-1}}(\z) \, .
    \]
\end{theo}
\begin{proof}
    We observe that
    \[
        F_g(\tfrac{1}{\z}) \ = \
        \sum_{\Lbot \prec b_1 \prec b_2 \prec \cdots \prec b_k \prec \Ltop}
        g(\Lbot, b_1) \frac{-1}{1 - z_{b_1}}
        g(b_1, b_2) \frac{-1}{1 - z_{b_2}}
        \cdots
        g(b_{k-1}, b_k)
        \frac{-1}{1 - z_{b_k}}
        \, .
    \]
    The coefficient $g(\Lbot, b_1) g(b_1, b_2) \cdots g(b_{k-1}, b_k)$ now contributes to every
    multichain supported on a subset of $\{ b_1, b_2 , \dots, b_k\}$. Rewriting, this is the same
    as
    \[
        F_g(\tfrac{1}{\z}) \ = \
        \sum_{\Lbot \prec a_1 \prec a_2 \prec \cdots \prec a_l \prec \Ltop}
        h(\Lbot, a_1) \frac{z_{a_1}}{1 - z_{a_1}}
        h(a_1, a_2) \frac{z_{a_2}}{1 - z_{a_2}}
        \cdots
        h(a_{l-1}, a_l)
        \frac{z_{a_l}}{1 - z_{a_l}} \, ,
    \]
    where for $u \prec v$
    \begin{align*}
        h(u,v) \ \defeq& \
        \sum_{u \prec b_1 \prec b_2 \prec \cdots \prec b_k \prec v}
        (-1)^k
        g(u,b_1) g(b_1,b_2) \cdots g(b_k,v)\\
        \ =& \ \sum_{k \ge 0} (-1)^k (g-\delta)^k(u,v) \ = \ g^{-1}(u,v) \, .
        \qedhere
    \end{align*}
\end{proof}

The above computation is reminiscent of calculation of the antipode applied to the quasisymmetric
function associated to a graded poset in Ehrenborg~\cite{Ehrenborg-hopf}. Applying this statement
to a pair $\alpha, \beta$ of complementary angles allows us to directly relate interior and
exterior flag-angles:

\begin{coro}\label{cor:int_to_ext}
    Let $\CA$ be a cone angle with complementary cone angle $\CAb$.  For every
    $d$-polytope $P$, the interior and exterior flag angle vectors are related
    via
    \[
        \sum_{S} (-1)^{d - t} \aInt_S(P) \prod_{i \in S} x_i
        \ = \ \sum_{S} \bExt_S(\poset) \prod_{i \in S} -(x_i+1)
        \ \in \ \C[x_1,\dots,x_d] \, ,
    \]
    where the sums are over all $S \subseteq [0, d-1]$ and $t = \min(S \cup \{d\})$.
\end{coro}
\begin{proof}
    Let $\Lfaces = \Lfaces(P)$ and
    $x_i \defeq \frac{q_i}{1 - q_i} \in R$. Then
    $q_i = \frac{x_i}{x_i + 1}$ and
    $\frac{-1}{1 - q_i} = -x_i-1$. Using Theorem~\ref{thm:unipot}, we compute
    \begin{align*}
      \sum_{S} (-1)^{d - t} \aInt_S(P) \prod_{i \in S} x_i 
      \ &= \ \sum_{S} \aInt'_S(P) \prod_{i \in S} x_i
      \  = \ \sum_{S} \aInt'_S(P) \prod_{i \in S} \frac{q_i}{1-q_i}\\
      \ &= \ F_{\aInt'}(z_a = q_{\rk(a)} : a \in \Lfaces)
      \  = \ F_{\bExt}(z_a = q_{\rk(a)}^{-1} : a \in \Lfaces)\\
      \ &= \ \sum_{S} \bExt_S(P) \prod_{i \in S} \frac{-1}{1-q_i}
      \  = \ \sum_{S} \bExt_S(P) \prod_{i \in S} -(x_i+1)\,,
    \end{align*}
    where each sum ranges over all $S \subseteq [0, d-1]$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:flag_rels}]
    Propositions~\ref{prop:flag_ext_rels} and~\ref{prop:flag_int_rels} yield
    that the linear relations given in Theorem~\ref{thm:flag_rels} hold. In
    particular, this shows that the dimensions of the affine hulls of
    interior/exterior flag-angles is at most $2^{d-1}-1$.

    From Theorem~\ref{thm:flag_whitney}, we infer that
    \begin{align*}
        \aff \{\faExt(P) : \text{$P$ $d$-polytope} \}
        \ &\supseteq \ \aff \{\faExt(Z) : \text{$Z$ $d$-zonotope} \}\\
        \ &= \ \aff \{\FW(\Lflats(Z)^\op) : \text{$Z$ $d$-zonotope} \} \, .
    \end{align*}
    Theorem~\ref{thm:zono_span}, that we will prove in the next section, shows that the dimension
    of the affine hull of flag-vectors of $\Lflats(Z)$ where $Z$ ranges over all $d$-dimensional
    zonotopes is of dimension $2^{d-1}-1$. This proves the claim for exterior flag-angle vectors.

    The same reasoning applies to the interior flag-angle vectors and it suffices to determine the
    affine span of $(w_S(\Lflats(Z)^\op))_S$ for $d$-dimensional zonotopes $Z$. Analogously to
    Corollary~\ref{cor:int_to_ext}, Theorem~\ref{thm:unipot} implies that the spaces of
    flag-Whitney numbers of the first and of the second kind spanned by posets of rank $d+1$ are
    linearly isomorphic, which completes the proof.
\end{proof}

\section{Flag-Whitney numbers and zonotopes}\label{sec:flag-whitney}

Let $\poset$ be a graded poset with $\Lbot$ and $\Ltop$ of rank $d+1$. It was shown by Billera and
Hetyei~\cite{BilleraHetyei} that flag-vectors of general graded posets do not satisfy any
nontrivial linear relation. That is
\[
    \dim \aff \{ \FW(\poset) \in \R^{2^d} \ : \ \poset \text{ graded poset of
        rank $d+1$} \} \ = \ 2^{d} - 1 \, .
\]
The only linear relation is given by $W_\emptyset (\poset)= 1$.

In light of Theorem~\ref{thm:flag_whitney}, we can complete the proof of
Theorem~\ref{thm:flag_rels} for exterior flag-angles by proving the following refinement of the
result of Billera and Hetyei.

\begin{theo}\label{thm:zono_span}
    The flag-vectors of lattices of flats of $(d+1)$-dimensional zonotopes
    span the flag-vectors of rank $d+1$ posets. That is,
    \[
        \dim\aff\{\FW(\Lflats(Z)) \ : \ Z \text{ zonotope of dimension $d+1$}\}
        \ = \ 2^d - 1 \, .
    \]
\end{theo}

The result is analogous to that of Billera--Ehrenborg--Readdy~\cite{BER} where it is shown that the
flag-vectors of face lattices of zonotopes span the space of flag-vectors of Eulerian posets. For
the proof of Theorem~\ref{thm:zono_span}, we will employ the coalgebra techniques developed
in~\cite{BER}.

\newcommand{\A}{\mathcal{A}} 
Let $\A = k\langle a,b \rangle$ be the polynomial ring in noncommuting
variables $a$ and $b$. This is a graded algebra $\A = \bigoplus_{d\geq 0}
\A_d$ and a basis for $\A_d$ is given by $\{a-b,b\}^d$. The \Def{ab-Index} of
a graded poset $\poset$ of rank $d+1$ is given by
\[
    \Psi(\poset) \ = \ \sum_{S \subseteq [d]} W_S(\poset) x(S)
\]
where $x(S) = x_1x_2\dots x_d \in \{a-b,b\}^d$ with $x_i = b$ if and only if $i \in S$.

Following~\cite{BER}, we consider two natural operations on zonotopes: If $Z \subset \R^d$ is a
zonotope, then $\abE(Z) \defeq Z \times [0,1] \subset \R^{d+1}$ is a zonotope of dimension $\dim Z
+ 1$. This is clearly a combinatorial construction and for the lattice of flats $\Lflats =
\Lflats(Z)$, we note that
\[
    \abE(\Lflats) \ \defeq \ \Lflats(\abE(Z)) \ = \ \Lflats \times C_1 \, ,
\]
where $C_1 = \{ \Lbot \prec \Ltop \}$ is the chain on $2$ elements. A vector $u \in \affL(Z)$ is
  in general position with respect to $Z$ if $u$ is not parallel to any face of $Z$. It can be
  shown (see~\cite[Ch.~7]{White-matroids}) that the lattice of flats of $\abM(Z) \defeq Z + [0,u]$
  is independent of the choice of $u$ and given by
\[
    \abM(\Lflats) \ \defeq \ \Lflats(\abM(Z)) \ = \ \abE(\Lflats) \setminus
    \{ x \in \abE(\Lflats) : \rk(x) = d+1 \} \, .
\]
Let $\abP(Z)$ be the orthogonal projection of $\abM(Z)$ onto the hyperplane $u^\perp$. This again
  is a combinatorial operation and $\abP(\Lflats) \defeq \Lflats(\abP(Z))$ is obtained from
  $\Lflats$ by deleting the coatoms, that is, elements of $\rk(\Lflats) - 1$.

In order to determine the effect on the ab-index, we introduce derivations ${\abR, \abR': \A \to
\A}$ defined on the variables by $\abR(a) \defeq \abR(b) \defeq ab$ and $\abR'(a) \defeq R'(b)
\defeq ba$ and linearly extended via
\[
    \abR(xy)  \  \defeq \ \abR(x)y + x\abR(y)  \qquad
    \abR'(xy)  \ \defeq \ \abR'(x)y + x\abR'(y)
\]
for monomials $x,y$. Note that both derivations are homogeneous and map $\A_d$ into $\A_{d+1}$.
  We also define linear maps $\abE,\abM,\abP: \A \to \A$ on monomials $x$ by
\[
    \begin{array}{l@{\ \ \defeq \ \ }l@{\qquad}l@{\ \ \defeq \ \ }l}
        \abP(xa) & x & \abE(x) & xa + bx + \abR(x)  \\
        \abP(xb) & 0 & \abM(x) & \abP(\abE(x)) \, . \\
    \end{array}
\]
In particular, we have
\[
    \abM(xa) \ = \ xa + bx + \abR(x) \ = \ \abE(x) \qquad \text{ and } \qquad
    \abM(xb) \ = \
    xb \, .
\]

The following result can be easily obtained by inspecting chains.
\begin{lemm}
    Let $Z$ be zonotope and $\Lflats = \Lflats(Z)$ its lattice of flats. Then
    \begin{align*}
        \Psi(\abE(\Lflats))    \ &= \ \abE(\Psi(\Lflats)) \ = \
        \Psi(\Lflats)b + a\Psi(\Lflats) + \abR'(\Psi(\Lflats)),\\
        \Psi(\abP(\Lflats))  \ &= \ \abP(\Psi(\Lflats)), \text{ and}\\
        \Psi(\abM(\Lflats))  \ &= \ \abP(\abE(\Psi(\Lflats))) \, .
    \end{align*}
\end{lemm}

\begin{proof}[Proof of Theorem~\ref{thm:zono_span}]
    \newcommand{\ZZ}{\mathcal{Z}}
    For $d \ge 0$ let
    \[
        \ZZ_d \ \defeq \
        \mathrm{span} \{ \Psi(\Lflats(Z)) : Z \text{ zonotope of dimension $d+1$} \} \
        \subseteq \ \A_d \, .
    \]
    We show by induction on $d$ that $\ZZ_d = \A_d$. For $d=1$ this is clearly true. Assume that
    $\ZZ_d = \A_d$. The key observation is that if $x$ is any monomial in $\A_d = \ZZ_d$, then also
    $\abM(x) \in \ZZ_{d}$ and $\abE(x) \in \ZZ_{d+1}$.

    \begin{enumerate}[\rm (i)]
        \item $xba \in \ZZ_{d+1}$ for all $x \in \ZZ_{d-1}$:
            \[
                2 \abE(xb) - \abM(\abE(xb)) \ = \ xba  \, .
            \]

        \item $xab \in \ZZ_{d+1}$ for all $x \in \ZZ_{d-1}$:
            \[
                \abM(\abE(xa)) - \abE(xa + bx + \abR(x)) \ = \ xab \, .
            \]

        \item $xba^n \in \ZZ_{d+1}$ for all $x \in \ZZ_{d - n}, n = 1,
            \dots, d - 1$:\\
            For $n = 1$ this is just (i). We may assume that the claim holds for all
            values $< n+1$ and compute
            \[
                \abE(xba^n)  \ = \ xba^{n+1} + xba^nb + \sum_{i=1}^{n}x_iba^{i}
            \]
            for some $x_i \in A_{d-i-1}$. Since $x_iba^i \in \ZZ_{d+1}$ by induction and $xba^nb
            \in \ZZ_{d+1}$ by (ii), we see that $xba^{n+1} \in \ZZ_{d+1}$.

        \item $xab^n \in \ZZ_{d+1}$ for all $x \in \ZZ_{d - n}, n = 1, \dots, d -
            1$:\\
            For $n = 1$ this is just (ii). Assume the statement holds for all values
            $<n+1$:
            \[
                \abE(xab^n) \ = \ xab^{n+1} + xab^na + \sum_{i=1}^{n}x_iab^{i}
            \]
            for some $x_i \in A_{d-i-1}$. Since $x_iab^i \in \ZZ_{d+1}$ by induction and $xab^na
            \in \ZZ_{d+1}$ by (i), we see that $xab^{n+1} \in \ZZ_{d+1}$.
    \end{enumerate}

    Since every monomial in $\A_{d+1}$ which contains at least one $a$ and $b$ is of either the
    form $xab^n$ or $xba^n$, we see that it remains to show that $a^{d+1}$ and $b^{d+1}$ are in
    $\ZZ_{d+1}$ as well. For that we compute
    \begin{align*}
      \abE(a^d) &\ = \ a^{d+1} + ba^d + \abR(a^d)\\
      \abE(b^d) &\ = \ b^da + b^{d+1} + \abR(b^d)
    \end{align*}
    Since $ba^d, b^da, \abR(a^d), \abR(b^d) \in \ZZ_{d+1}$, this finishes the proof.
\end{proof}

In fact we have proven the following statement:
\begin{coro}
    For $d \ge 0$, a vector space basis of $\A_d$ is given by
    \[
        \bigl\{ \Phi(\Lflats( \sigma [0,1] )) : \sigma \in
        \{\abE,\abM\circ\abE\}^d \bigr\} \, .
    \]
\end{coro}
\begin{proof}
    In the proof of Theorem~\ref{thm:zono_span} we only needed elements of the
    form $E(x)$ and $M(E(x))$, $x \in \A_d$, to span $\A_{d+1}$, thus the
    assertion follows by induction.
\end{proof}

\section{Spherical intrinsic volumes and Grassmann angles}
\label{sec:spherical_intrinsic_volumes}
We have already seen spherical intrinsic volumes in Section~\ref{sec:incalg},
but as a further introduction and motivation to flag-angles, we will give a
slightly more general account here. Moreover, we will see how interior and
exterior flag-angles are connected. A second goal of this chapter is to show how
we can adapt and generalize many results by Gr\"{u}nbaum~\cite{Grunbaum-grassmann}
to a more general version of Grassmann angles.

For convenience, we want to define for a cone $C \subset \R^d$ the \Def{completion} as ${\cpl C
\defeq C + \affL(C)^\perp}$. Recall that $\SCA$ denotes the spherical volume. For $0 \leq r \leq
d$, the \Defn{$r$-th spherical intrinsic volume} $\SCA^r$ of a cone $C \in \Cones^d$ is defined as:
\[
    \SCA^r(C) \ \defeq \ \sum_{\substack{F \subseteq C \text{ face} \\ \dim F = r}}
    \SCA(\cpl F) \cdot \SCA( \Ocone{F}{C} ) \, ,
\]
where $\Ocone{F}{C} = \cpl \Ncone{F}{C} = \Ncone{F}{C} + \affL(F)$. Note that $\SCA^d(C) =
  \SCA(C)$ and ${\SCA^i(C) = \SCA^{d-i}(C^\polar)}$ for all $i$. The name \emph{spherical}
  intrinsic volume stems from the similarity to a formula for the usual intrinsic volumes $V^r$ of
  a polytope $P$
\[
    V^r(P) = \sum_{\dim F = r} \vol_r(F) \cdot \SCA(\Ocone{F}{P})\,,
\]
where $\vol_r$ denotes the usual $r$-dimensional volume; see~\cite[Section 4.2]{Schneider}. The
  intersection of a face $F \subseteq C$ with the unit sphere $\Sphere^d$ is a spherical polytope
  and $\SCA(\cpl F)$ is the normalized spherical volume.

We would like to replace $\SCA$ in the definition of $\SCA^r$ with more general valuations. In
fact, we could replace both occurrences of $\SCA$ with different valuations. Let $\alpha$, $\beta$
be cone angles. For $0 \leq r \leq d$, we define the \Defn{generalized spherical intrinsic volume}
$\xi^r = \xi^r(\alpha, \beta) : \Cones^d \to \R$ by
\begin{equation}
    \xi^r(C) \ \defeq \ 
    \sum_{\substack{F \subseteq C \text{ face} \\ \dim F = r}} \alpha(\cpl F) \cdot
    \beta(\Ocone{F}{C})\,,
    \label{eqn:gen_sph_int_vol}
\end{equation}
It is well known that $\SCA^r$ is a valuation for all $r$;
see~\cite[Lemma~2.3.2]{Schneider_ConvexCones} for an elementary proof. This remains true for the
generalized spherical intrinsic volumes:

\begin{theo}\label{thm:generalized_spherical_intrinsic_volume_is_valuation}
    Let $\alpha, \beta : \Cones^d \to \R$ be cone angles. Then $\xi^r(\alpha,
    \beta)$ is a valuation for $0 \leq r \leq d$.
\end{theo}

Note that $\xi(\alpha, \beta)$ is not a simple valuation: if $C$ is a linear subspace of dimension
$r < d$, then $\xi^r(\alpha,\beta)(C) = \alpha(\R^d)\beta(\R^d) = 1$. Furthermore, recall that we
can view the associated interior and exterior cone angles $\aInt$ and $\bExt$ as elements in the
incidence algebra of the face lattice $\Lfaces(C)$. This allows the interpretation
\[
    \xi^r(C) \ = \ \big(\aInt \ast_r \bExt\big)(F, C) \, ,
\]
where $F$ is the smallest non-empty face of $C$, that is, the largest linear subspace contained
  in $C$. Theorem~\ref{thm:generalized_spherical_intrinsic_volume_is_valuation} suggests that
  higher products such as $(\aInt_1 \ast_r \aExt_2) \ast_s \aExt_3$ are valuations as well. This,
  unfortunately, is not the case as the simplicity of $\alpha$ and $\beta$ is essential in the
  proof of Theorem~\ref{thm:generalized_spherical_intrinsic_volume_is_valuation}.

\begin{proof}[Proof of
        Theorem~\ref{thm:generalized_spherical_intrinsic_volume_is_valuation}]
    By~\cite{Sallee} it suffices to show that $\xi = \xi^r(\alpha, \beta)$ is a
    \emph{weak} valuation: For every cone $C \subset \R^d$ and $H$ a linear
    hyperplane we need to show that
    \begin{equation}\label{eqn:xi_valuation}
        \xi(C)  \ = \ \xi(C \cap H^\le) + \xi(C \cap H^\ge) - \xi(C \cap H)  \,
        .
    \end{equation}
    It is sufficient to assume that $C \not\subseteq H$ and that $H$ meets the relative interior of
    $C$. Then the cones $C^\le \defeq C \cap H^\le$ and $C^\ge \defeq C \cap H^\ge$ are of the same
    dimension as $C$ and $C^= \defeq C \cap H$ is of dimension $\dim(C) - 1$.

    To show~\eqref{eqn:xi_valuation}, we need to consider all $r$-faces of $C^\leq$, $C^\geq$, and
    $C^=$. These faces are either faces of $C$ or are obtained by intersecting faces of $C$ with
    $H^\leq$, $H^\geq$ or $H$ in the following ways. Let $F$ be an $r$-face of $C$.
    \begin{enumerate}[{Case} 1.]
        \item If $\relint(F) \cap H = \emptyset$, then $F$ is contained in $H^\leq$ or $H^\geq$ and
            $F$ is an $r$-face of $C^\leq$ or $C^\geq$, respectively.
        \item If $F \subseteq H$, then $F$ is an $r$-face of $C^\leq$, $C^\geq$, and $C^=$.
        \item If $H$ intersects $\relint(F)$ in a proper subset, then $F \cap H^\leq$ is an
            $r$-face of $C^\leq$ and $F \cap H^\geq$ is an $r$-face of $C^\geq$. Further more $F
            \cap H$ is an $(r-1)$-face of $C^\leq$, $C^\geq$ and $C^=$.
    \end{enumerate}
    Furthermore, let $G \subseteq C$ be an $(r+1)$-face.
    \begin{enumerate}[{Case} 4.]
        \item If $\relint(G) \cap H \neq \emptyset$, then $G \cap H$ is an $r$-face of $C^\leq$,
            $C^\geq$, and $C^=$.
    \end{enumerate}

    We consider the contributions of each case to~\eqref{eqn:xi_valuation} separately:

    Case 1. Without loss of generality we can assume that $F \subseteq H^\le$ so that $F$ is a face
    of $C^\le$ as well. In this case $\Ocone{F}{C} = \Ocone{F}{C^\le}$ and thus $F$ gives the same
    contribution to $\xi(C)$ and $\xi(C^\le)$ and none to $\xi(C^\ge)$ and $\xi(C^=)$.

    Case 2. As $F$ is a face of all four cones, we have $\Ocone{F}{C} = \Ocone{F}{C^{\le}} \cup
    \Ocone{F}{C^{\ge}}$ with $\Ocone{F}{C^{\le}} \cap \Ocone{F}{C^{\ge}} = \Ocone{F}{C^=}$. The
    contribution on the right-hand side is then
    \[
        \alpha(\cpl F)( \beta(\Ocone{F}{C^\le}) +
        \beta(\Ocone{F}{C^\ge}) -
        \beta(\Ocone{F}{C^=})) \, ,
    \]
    which equals to $\alpha(\cpl F)\beta(\Ocone{F}{C})$ as $\beta$ is a valuation.

    Case 3. Set $F^\le = F \cap H^\le$ and $F^\ge = F \cap H^\ge$, which are faces of $C^\le$ and
    $C^\ge$, respectively. Since the normal cone is polar to the tangent cone and the tangent cone
    of a face $F$ is determined by any neighborhood of a point $q \in \relint(F)$, we have
    $\Ocone{F}{C} = \Ocone{F^\le}{C^\le} = \Ocone{F^\ge}{C^\ge}$. The contribution on the
    right-hand side is therefore
    \[
        (\alpha(\cpl F^\le) + \alpha(\cpl F^\ge)) \beta(\Ocone{F}{C})
    \]
    which is precisely $\alpha(\cpl F) \beta(\Ocone{F}{C})$ since $\alpha$ is a simple valuation.

    Case 4. Here $F^= = F \cap H$ is a common face of $C^\le$, $C^\ge$, and $C^=$. Since
    ${\Ocone{F^=}{C^=} = \Ocone{F^=}{C^{\le}} \cup \Ocone{F^=}{C^{\ge}}}$ and $\beta$ is a simple
    valuation, the contribution to the right-hand side is $0$.
\end{proof}

Since $\xi^r = \xi^r(\alpha, \beta)$ is a valuation, we immediately obtain the following from the
Brianchon-Gram relation. For a $d$-polytope $P$ and a face $F \subseteq P$ define
${\widehat{\xi}^r_i(P) \defeq \sum_{F} \xi^r(T_FP)}$ where the sum is over all $i$-faces $F$ of
$P$. Applying $\xi^r$ to the general form of the Brianchon--Gram relation~\eqref{eqn:gen_BR}, we
obtain
\begin{coro}
    Let $\alpha, \beta : \Cones^d \to \R$ be cone angles and
    $\xi^r = \xi^r(\alpha, \beta)$ for some $0 \leq r \leq d$. Then
    \[
        \widehat{\xi}^r_0(P) - \widehat{\xi}^r_1(P) + \widehat{\xi}^r_2(P) -
        \dots + (-1)^{\dim d} \cdot \widehat{\xi}^r_d(P)
        \ = \
        \begin{cases}1 & \text{ if } r = 0\\0& \text{ if } r > 0\end{cases} \, .
    \]
\end{coro}

If $\beta$ is the (unique) cone angle complementary to $\alpha$, we will simplify the notation and
write $\xi^r(\alpha) \defeq \xi^r(\alpha,\beta)$. When unraveling the definition of complementary
angles, we obtain an equation sometimes called the Gauss--Bonnet Theorem for polyhedral
cones~\cite{AL}. Let $\chi$ be the Euler characteristic on cones with $\chi(D) = 0$ if $D$ is not a
linear subspace and $\chi(D) = (-1)^{\dim D}$ otherwise.
\begin{lemm}\label{lem:xi_zero_one}
    Let $\alpha$ be a cone angle and $C \in \Cones^d$. Then
    \[
        \sum_{r=0}^{d} (-1)^r \xi^r(\alpha)(C) \ = \ \chi(C) \, .
    \]
\end{lemm}

\begin{rema}
    The classical spherical intrinsic volumes furthermore sum to
    $1$. This is not necessarily true for generalized spherical
    intrinsic volumes, but one can show using some calculation
    involving Lemma~\ref{lem:AS} that this holds for $\xi(\alpha)$
    when we additionally assume that the cone angle $\alpha$ is
    \emph{even}, that is, that $\alpha(C) = \alpha(-C)$ for all
    polyhedral cones $C \subseteq \R^d$.
\end{rema}

\newcommand{\Gr}{\operatorname{Gr}}
\newcommand{\gr}{\kappa}
We will now express the spherical intrinsic volumes in a different basis.
Using integral geometry, we can relate the usual spherical intrinsic volumes
$\nu^r$ to certain integrals over the \Def{Grassmannian} $\Gr^{r, d}$ of
$r$-dimensional linear subspaces in $\R^d$. The \Def{Haar measure} $\mu$
is the unique $O(d)$-invariant measure  on $\Gr^{r, d}$ such that ${\mu(\Gr^{r, d}) = 1}$.
The \Defn{$r$-th Grassmann-angle} of a cone $C \subset \R^d$
\[
    \gr^r(C) \ \defeq \ \mu(\{L \in \Gr^{r, d} : L \cap C = \{0\}\})
\]
was introduced by Gr\"{u}nbaum in~\cite{Grunbaum-grassmann} as a generalization of interior and
  exterior angles. Indeed $2 \SCA(C) = 1 - \gr^1(C)$ and $2 \SCA(C^\polar) = \gr^{d-1}(C)$. To
  generalize the Grassmann angles to arbitrary valuations we need to shift our point of view. For a
  fixed pointed cone $C \in \Cones^d$, define the $\mu$-measurable function $\eps_C : \Gr^{r,d} \to
  \{0,1\}$ with $\eps_C(L) \defeq 1$ if $C \cap L = \{0\}$ and $0$ otherwise. The Grassmann-angle
  can now be expressed as the integral over $\eps_C$
\[
    \gr^r(C) \ = \ \int_{\Gr^{r, d}} \eps_C(L)\,d\mu(L)
\]
Recall the kinematic formulas for cones.

\begin{theo}[{\cite[Theorem~5.1]{AL}}]
    Let $C \subseteq \R^d$ be a polyhedral cone. Then for $0 \leq r \leq d$ and $1 \leq k \leq d$:
    \begin{align*}
      \int_{\Gr^{r, d}} \nu^k(C \cap L) d\mu(L) \ &= \ \nu^{k + d - r}(C)\,, &
      \int_{\Gr^{r, d}} \nu^0(C \cap L) d\mu(L) \ &= \ \sum_{j=0}^{d-r} \nu^{j}(C)\,.
    \end{align*}
\end{theo}

For a fixed cone, we have almost surely $\eps_C(L) = \chi(C \cap L)$. From
Lemma~\ref{lem:xi_zero_one} we get
\[
    \chi(C) = \sum_{i = 0}^d (-1)^i \nu^i(C) \, ,
\]
and we compute
\begin{align*}
  \kappa^r(C)
  \ &= \ \int_{\Gr^{r,d}} \eps_C(L) d\mu(L) \ = \ \int_{\Gr^{r,d}} \chi(C \cap L) d\mu(L) \\
    &= \  \sum_{i = 0}^d (-1)^i \int_{\Gr^{r,d}} \nu^i(C \cap L) d\mu(L) 
    \ = \ \sum_{j = 0}^{d-r}\nu^j(C) + \sum_{i = 1}^d (-1)^i \nu^{i+d-r}(C) \\
    &= \ \sum_{j = 0}^{d-r}\nu^j(C) + \sum_{i = d-r+1}^d (-1)^{i + d - r} \nu^{i}(C)\,.
\end{align*}

This is a slight variation of the usual Crofton-formulas, which better serves our purposes. We
refer to~\cite{AL} for further details. It is not hard to see that spherical intrinsic volumes and
Grassmann-angles encode the same quantities in a different basis, and conversely we obtain the
intrinsic volumes from the Grassmann-angles as follows:
\[
    \nu^r = \tfrac{1}{2} \big(\gr^{d-r-1} - \gr^{d-r+1}\big)
\]
for $r = 1, \dots, d-1$ as well as $\nu^0 = \frac{1}{2} (\gr^d + \gr^{d-1})$ and $\nu^d =
  \frac{1}{2} (\gr^0 - \gr^1)$.

Using the generalized spherical volumes allows us to give a generalization of the Grassmann angles,
too, by taking the Crofton-formulas as a definition. Thus we define for any two cone angles
$\alpha, \beta : \Cones^d \to \R$ the \Defn{generalized $r$-th Grassmann-angle}
\[
    \gr^r(\alpha, \beta) \defeq \sum_{j = 0}^{d - r} \xi^j(\alpha, \beta) +
    \sum_{i = d-r+1}^{d} (-1)^{i-d+r} \xi^i(\alpha, \beta)\,.
\]
we will simplify write $\xi^r(\alpha) = \xi^r(\alpha, \beta)$ and $\gr^r(\alpha) = \gr^r(\alpha,
  \beta)$ if $\beta$ is the (unique) complementary angle to $\alpha$.

As a corollary of Theorem~\ref{thm:generalized_spherical_intrinsic_volume_is_valuation}, we have:
\begin{coro}
    Every generalized Grassmann-angle is a valuation.
\end{coro}

From this observation we can draw short proofs for most of the results in Gr\"{u}nbaum's original paper
on Grassmann angles~\cite{Grunbaum-grassmann} where at the same time we replace the usual
Grassmann-angle with our generalized notion $\gr^r = \gr^r(\alpha)$.
\newcommand{\grInt}{\widehat{\gr}} Let us denote by $\grInt_i^r(P)$ the sums of all $r$-th
generalized Grassmann angles of the $i$-faces of a $d$-polytope $P \subseteq \R^d$, that is
\[
    \grInt_i^r(P) \ \defeq \ \sum_{F} \gr^r(T_FP)\,.
\]
We have:

\begin{coro}[Generalization of Gr\"{u}nbaum~{\cite[Theorem~3.3]{Grunbaum-grassmann}}]
    Let $P \subset \R^d$ be a $d$-polytope and
    $\gr^r = \gr^r(\alpha)$ for a cone angle
    $\alpha$ and $0 \leq r \leq d$. Then
    \[
        \sum_{i = 0}^{d - r} (-1)^{i} \cdot \grInt^r_i(P) \ = \ 1\,.
    \]
\end{coro}
\begin{proof}
    Since $\xi^0(\{0\}) = 1$ and
    $\xi^r(\{0\}) = 0$ for all $1 \leq r \leq d$, we have
    $\gr^r(\{0\}) = 1$ for all $0 \leq r \leq d$. Applying $\gr^r$
    to both sides of~\eqref{eqn:gen_BR} yields
    \begin{equation}
       \gr^r(\{0\}) + \sum_{F} (-1)^{\dim F + 1} \gr^r(T_FP) + (-1)^{d+1} \gr^r(\R^d) = 0\,.
       \label{eqn:grassmann_BR}
   \end{equation}
    If $F \subseteq P$ is a face with $\dim F > d-r$, then $\xi^j(T_FP) = 0$ for all $j \leq d-r$,
    as the smallest face of $T_FP$ has dimension $\dim F$ and thus the sum in~\eqref{eqn:gen_sph_int_vol} is empty. Thus, by Lemma~\ref{lem:xi_zero_one}
    \begin{align*}
     \gr^r(T_FP)
     \ &= \ \sum_{j = 0}^{d-r} \xi^j(T_FP) + \sum_{i = d-r+1}^d (-1)^{i-d+r} \xi^i(T_FP)\\
     \ &= \ (-1)^{d-r} \sum_{i = 0}^{d} (-1)^{i} \xi^i(T_FP) \ = \ 0\,.
   \end{align*}
    With that, we obtain the claim by rearranging~\eqref{eqn:grassmann_BR}.
\end{proof}

In a similar fashion, most of the results in~\cite{Grunbaum-grassmann} can be shown for generalized
Grassmann angles. For example,~\cite[Theorem~3.5]{Grunbaum-grassmann} follows from an application
$\kappa^r$ to Lemma~\ref{lem:AS}.

\longthanks{Research that led to this paper was supported by the
DFG-Collabora\-tive Research Center, TRR 109 ``Discretization in Geometry and
Dynamics'' and by the National Science Foundation under Grant No.~DMS-1440140
while the authors were at the Mathematical Sciences Research Institute in
Berkeley, California, during the Fall 2017 semester on \emph{Geometric and
    Topological Combinatorics}.  S.~Backman was also supported by a Zuckerman STEM
Postdoctoral Scholarship. We thank Marge Bayer, Curtis Greene, Carly Klivans,
and Richard Ehrenborg for helpful discussions.}

\bibliographystyle{mersenne-plain-nobysame}
\bibliography{ALCO_Sanyal_1276}
\end{document}
