Union of Translates of a Convex Body

Overview & Synopsis

Let $K\subset\mathbb{R}^d$ be a strictly convex body and let $Z\subset\mathbb{R}^d$ be a finite set of sites in general position. We consider the problem of calculating the volume $μ_d$ of $\bigcup_{z\in Z}(z-K)$. Let $V_z$ be the Voronoi cell of $z\in Z$ with respect to the convex distance function induced by $K$, and let $N$ be the nerve of the Voronoi cells. When $K$ is a ball, a classical result reduces the inclusion-exclusion formula to a sum over $N$: $μ_d(\bigcup_{z\in Z}(z-K))=\sum_{J\in N}(-1)^{|J|+1}μ_d(\bigcap_{z\in J}(z-K))$. When may the hypothesis that $K$ is a ball be dropped? We show that for $d\leq 2$ the reduced formula holds for every strictly convex $K$, while for each $d\geq 3$ there is a strictly convex $K$ for which it fails. For $x\in\mathbb{R}^d$, let $χ_x$ be the Euler characteristic of the subcomplex of $N$ induced by the sites lying in $K+x$. A sufficient condition for the reduced formula to hold is $χ_x\equiv 1$ on the union. When $d=3$ and $K$ is $C^4$, we show that $χ_x\equiv 1$ on the union for all $Z$ in general position if and only if $K$ is an ellipsoid. The proof relies on the Fubini-Pick affine invariant.