# Canonical decomposition of a difference of convex sets

## TL;DR

This paper introduces a canonical convex decomposition for the difference of two convex sets in a lattice setting and relates the volume of the pieces to intersection numbers of toric b-divisors, advancing geometric understanding.

## Contribution

It provides a canonical decomposition of the difference of convex sets and links the volume of parts to intersection theory of toric b-divisors, a novel geometric interpretation.

## Key findings

- Canonical convex decomposition of the difference of convex sets.
- Geometric interpretation of volumes via intersection numbers.
- Enhanced understanding of convex set differences in lattice geometry.

## 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 $K_1 \subseteq K_2 \subseteq M_{\R} \coloneqq M \otimes_{\Z} \R$, 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.