Canonical decomposition of a difference of convex sets

Abstract

Let $N$ be a lattice of rank $n$ and let $M=N^{\vee }$ be its dual lattice. In this note we show that given two compact, 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 $K_2\setminus K_1$ and we interpret the volume of the pieces geometrically in terms of intersection numbers of toric $b$-divisors.

20 pages