Isometric and affine copies of a set in volumetric Helly results

Abstract

We show that for any compact convex set K in R d and any finite family F of convex sets in R d , if the intersection of every sufficiently small subfamily of 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−ε), where ε 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 .