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

Let N be a lattice of rank n and let M=N be its dual lattice. In this article we show that given two closed, bounded, full-dimensional convex sets K 1 K 2 M :=M , there is a canonical convex decomposition of the difference K 2 int(K 1 ) and we interpret the volume of the pieces geometrically in terms of intersection numbers of toric b-divisors.

Received : 2018-03-24
Revised : 2018-09-10
Accepted : 2018-12-21
Published online : 2019-08-01
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},
     publisher = {MathOA foundation},
     volume = {2},
     number = {4},
     year = {2019},
     pages = {585-602},
     doi = {10.5802/alco.55},
     language = {en},
     url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_4_585_0}
}
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/item/ALCO_2019__2_4_585_0/

[1] Botero, A. M. Intersection theory of b-divisors in toric varieties, J. Algebraic Geom. (2018) (Accepted) | MR 3912060

[2] Burgos Gil, J. I.; Philippon, P.; Sombra, M. Arithmetic geometry of toric varieties. Metrics, measures and heights, Société Mathématique de France, Astérisque (2014) no. 360, vi+222 pages | MR 3222615 | Zbl 1311.14050

[3] Cox, D.; Little, J. B.; Schenck, H. Toric Varieties, Amer. Math. Soc., Graduate Studies in Mathematics, Volume 124 (2011) | MR 2810322 | Zbl 1223.14001

[4] Fulton, W. Introduction to Toric Varieties, Princeton Univ. Press (1993)

[5] Goodman, J. E.; Pach, J. Cell decomposition of polytopes by bending, Israel J. Math., Volume 64 (1988) no. 2, pp. 129-138 | MR 986142 | Zbl 0673.52005

[6] Griffiths, P.; Harris, J. Principles of Algebraic Geometry, Bull. Amer. Math. Soc., Volume 2 (1980) no. 1, pp. 197-200 | MR 1567210

[7] Hiriart-Urruty, J. B.; Lemaréchal, C. Fundamentals of convex analysis, Springer, Grundlehren Text Editions (2001) | Article | Zbl 0998.49001

[8] Kaveh, K.; Khovanskii, A. G. Convex bodies and algebraic equations on affine varieties (2008) (https://arxiv.org/abs/0804.4095v1 )

[9] Kaveh, K.; Khovanskii, A. G. Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), Volume 176 (2012) no. 2, pp. 925-978 | Article | MR 2950767 | Zbl 1270.14022

[10] Kaveh, K.; Khovanskii, A. G. Convex bodies and multiplicities of ideals, Proc. Steklov Inst. Math., Volume 286 (2014) no. 1, pp. 268-284 | Article | MR 3482603

[11] Lazarsfeld, R.; Mustaţă, M. Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), Volume 42 (2009) no. 5, pp. 783-835 | Article | MR 2571958 | Zbl 1182.14004

[12] Okounkov, A. Brunn–Minkowski inequality for multiplicities, Invent. Math., Volume 125 (1996) no. 3, pp. 405-411 | Article | MR 1400312 | Zbl 0893.52004

[13] Okounkov, A. Why would multiplicities be log-concave?, The orbit method in geometry and physics (Marseille, 2000), Birkhäuser, Boston (Progr. Math.) Volume 213 (2003), pp. 329-347 | Article | MR 1995384 | Zbl 1063.22024

[14] Rockafellar, R. T. Convex Analysis, Princeton Univ. Press (1970) | Article | Zbl 0229.90020

[15] Schneider, R. Convex bodies: The Brunn–Minkowski theory, Cambridge University Press, Encyclopedia Math. Appl., Volume 44 (1993) | MR 1216521 | Zbl 0798.52001