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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.
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
Full Text: DOI
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