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 $(1\varepsilon)$, where $\varepsilon$ 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 $P$.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 arXiv:
 arXiv:2010.04135
 Bibcode:
 2020arXiv201004135M
 Keywords:

 Mathematics  Metric Geometry;
 Computer Science  Computational Geometry;
 Mathematics  Combinatorics
 EPrint:
 10 pages, 2 figures