# ALGEBRAIC COMBINATORICS

Canonical decomposition of a difference of convex sets
Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 585-602.

Let $N$ be a lattice of rank $n$ and let $M={N}^{\vee }$ be its dual lattice. In this article we show that given two closed, bounded, full-dimensional convex sets ${K}_{1}\subseteq {K}_{2}\subseteq {M}_{ℝ}:=M{\otimes }_{ℤ}ℝ$, there is a canonical convex decomposition of the difference ${K}_{2}\setminus \text{int}\left({K}_{1}\right)$ and we interpret the volume of the pieces geometrically in terms of intersection numbers of toric $b$-divisors.

Revised:
Accepted:
Published online:
DOI: https://doi.org/10.5802/alco.55
Classification: 52A22,  14M25,  14C17
Keywords: convex geometry, toric geometry, intersection theory
@article{ALCO_2019__2_4_585_0,
author = {Botero, Ana M.},
title = {Canonical decomposition of a difference of convex sets},
journal = {Algebraic Combinatorics},
pages = {585--602},
publisher = {MathOA foundation},
volume = {2},
number = {4},
year = {2019},
doi = {10.5802/alco.55},
mrnumber = {3997512},
zbl = {1420.52005},
language = {en},
url = {https://alco.centre-mersenne.org/articles/10.5802/alco.55/}
}
Botero, Ana M. Canonical decomposition of a difference of convex sets. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 585-602. doi : 10.5802/alco.55. https://alco.centre-mersenne.org/articles/10.5802/alco.55/

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