Isometric and affine copies of a set in volumetric Helly results

TL;DR

This paper establishes volumetric Helly-type results for convex sets, showing that local isometric or affine copies of a set imply a scaled copy exists globally, with applications to approximation algorithms.

Contribution

It introduces volumetric Helly results for isometric and affine copies of convex sets, linking local intersections to global scaled copies, and develops efficient approximation algorithms.

Findings

Global scaled copies exist if small subfamilies contain isometric copies of a set

Results apply to affine copies of convex sets

Algorithms approximate largest inscribed copies with linear expected runtime

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 $P$.