Isometric and affine copies of a set in volumetric Helly results

Abstract

We show that for any compact convex set $K$ in ${\mathbb{R}}^d$ and any finite family $\mathcal{F}$ of convex sets in ${\mathbb{R}}^d$, if the intersection of every sufficiently small subfamily of $\mathcal{F}$ contains an isometric copy of $K$ of volume $1$, then the intersection of the whole family contains an isometric copy of $K$ scaled by a factor of $(1-\epsilon )$, where $\epsilon$ is positive and fixed in advance. Unless $K$ is very similar to a disk, the shrinking factor is unavoidable. We prove similar results for affine copies of $K$. We show how our results imply the existence of randomized algorithms that approximate the largest copy of $K$ that fits inside a given polytope $P$ whose expected runtime is linear on the number of facets of $P$.

10 pages, 2 figures