Union of Translates of a Convex Body

Abstract

We consider the problem of calculating the volume of the union of translates of a strictly convex body $K\subset\mathbb{R}^d$. When $K$ is a ball, it is possible to simplify the inclusion-exclusion formula by summing over only a small subset of all terms. This subset corresponds to intersecting cells in the Voronoi diagram of the ball centers. We prove that the same simplification is valid for every strictly convex $K$ if and only if $d\leq 2$. Moreover, in three dimensions and when $K$ is $C^4$, we use the Fubini-Pick affine invariant to show that the topological identity underlying this technique holds if and only if $K$ is an ellipsoid.