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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
##### Keywords:
convex geometry; toric geometry; intersection theory
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##### References:
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