×

zbMATH — the first resource for mathematics

Canonical decomposition of a difference of convex sets. (English) Zbl 1420.52005
Summary: 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 \(K_1 \subseteq K_2 \subseteq M_{\mathbb{R}} := M \otimes _{\mathbb{Z}} \mathbb{R}\), there is a canonical convex decomposition of the difference \(K_2 \setminus \text{int}(K_1)\) and we interpret the volume of the pieces geometrically in terms of intersection numbers of toric \(b\)-divisors.
MSC:
52A22 Random convex sets and integral geometry (aspects of convex geometry)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Botero, A. M., Intersection theory of \(b\)-divisors in toric varieties, J. Algebraic Geom., (2018)
[2] Burgos Gil, J. I.; Philippon, P.; Sombra, M., Arithmetic geometry of toric varieties. Metrics, measures and heights, 360, vi+222 p. pp., (2014), Société Mathématique de France · Zbl 1311.14050
[3] Cox, D.; Little, J. B.; Schenck, H., Toric Varieties, 124, (2011), Amer. Math. Soc. · Zbl 1223.14001
[4] Fulton, W., Introduction to Toric Varieties, (1993), Princeton Univ. Press
[5] Goodman, J. E.; Pach, J., Cell decomposition of polytopes by bending, Israel J. Math., 64, 2, 129-138, (1988) · Zbl 0673.52005
[6] Griffiths, P.; Harris, J., Principles of Algebraic Geometry, Bull. Amer. Math. Soc., 2, 1, 197-200, (1980)
[7] Hiriart-Urruty, J. B.; Lemaréchal, C., Fundamentals of convex analysis, (2001), Springer · Zbl 0998.49001
[8] Kaveh, K.; Khovanskii, A. G., Convex bodies and algebraic equations on affine varieties, (2008)
[9] Kaveh, K.; Khovanskii, A. G., Newton-Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Ann. of Math. (2), 176, 2, 925-978, (2012) · Zbl 1270.14022
[10] Kaveh, K.; Khovanskii, A. G., Convex bodies and multiplicities of ideals, Proc. Steklov Inst. Math., 286, 1, 268-284, (2014) · Zbl 1315.13013
[11] Lazarsfeld, R.; Mustaţă, M., Convex bodies associated to linear series, Ann. Sci. Éc. Norm. Supér. (4), 42, 5, 783-835, (2009) · Zbl 1182.14004
[12] Okounkov, A., Brunn-Minkowski inequality for multiplicities, Invent. Math., 125, 3, 405-411, (1996) · Zbl 0893.52004
[13] Okounkov, A., The orbit method in geometry and physics (Marseille, 2000), 213, Why would multiplicities be log-concave?, 329-347, (2003), Birkhäuser: Birkhäuser, Boston · Zbl 1063.22024
[14] Rockafellar, R. T., Convex Analysis, (1970), Princeton Univ. Press · Zbl 0229.90020
[15] Schneider, R., Convex bodies: The Brunn-Minkowski theory, 44, (1993), Cambridge University Press · Zbl 0798.52001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.