Geodesic
Canonical decomposition of a difference of convex sets
Botero, Ana M.1
1 University of Regensburg Dept. of mathematics Universitätsstr. 31 93053 Regensburg, Germany
Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 585-602.

Voir la notice de l'article provenant de la source Numdam

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 ℝ :=M⊗ ℤ ℝ, 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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/alco.55
Classification : 52A22, 14M25, 14C17
Keywords: convex geometry, toric geometry, intersection theory

Botero, Ana M. 1

1 University of Regensburg Dept. of mathematics Universitätsstr. 31 93053 Regensburg, Germany
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@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 = {http://geodesic.mathdoc.fr/articles/10.5802/alco.55/}
}
TY  - JOUR
AU  - Botero, Ana M.
TI  - Canonical decomposition of a difference of convex sets
JO  - Algebraic Combinatorics
PY  - 2019
SP  - 585
EP  - 602
VL  - 2
IS  - 4
PB  - MathOA foundation
UR  - http://geodesic.mathdoc.fr/articles/10.5802/alco.55/
DO  - 10.5802/alco.55
LA  - en
ID  - ALCO_2019__2_4_585_0
ER  - 
%0 Journal Article
%A Botero, Ana M.
%T Canonical decomposition of a difference of convex sets
%J Algebraic Combinatorics
%D 2019
%P 585-602
%V 2
%N 4
%I MathOA foundation
%U http://geodesic.mathdoc.fr/articles/10.5802/alco.55/
%R 10.5802/alco.55
%G en
%F ALCO_2019__2_4_585_0
Botero, Ana M. Canonical decomposition of a difference of convex sets. Algebraic Combinatorics, Tome 2 (2019) no. 4, pp. 585-602. doi : 10.5802/alco.55. http://geodesic.mathdoc.fr/articles/10.5802/alco.55/

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

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

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

[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 | Zbl | MR | DOI

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

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

[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 | Zbl | MR | DOI

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

[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 | Zbl | MR | DOI

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

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

[14] Rockafellar, R. T. Convex Analysis, Princeton Univ. Press, 1970 | DOI

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

Cité par Sources :