Homothetic covering of convex hulls of compact convex sets

TL;DR

This paper investigates how convex hulls of compact convex sets can be covered by smaller homothetic copies, providing bounds on the number needed and characterizing specific cases like parallelepipeds.

Contribution

It introduces estimations for covering functionals of convex hulls and establishes an upper bound of 8 for the covering number of certain 3D convex bodies, characterizing when this bound is tight.

Findings

The covering functional of convex hulls can be estimated based on the sets involved.

Any 3D convex body formed as the convex hull of two point-only sets can be covered by at most 8 smaller homothetic copies.

Parallelepipeds are uniquely characterized by requiring exactly 8 homothetic copies for coverage.

Abstract

Let $K$ be a compact convex set and $m$ be a positive integer. The covering functional of $K$ with respect to $m$ is the smallest $\lambda\in[0,1]$ such that $K$ can be covered by $m$ translates of $\lambda K$. Estimations of the covering functionals of convex hulls of two or more compact convex sets are presented. It is proved that, if a three-dimensional convex body $K$ is the convex hull of two compact convex sets having no interior points, then the least number $c(K)$ of smaller homothetic copies of $K$ needed to cover $K$ is not greater than $8$ and $c(K)=8$ if and only if $K$ is a parallelepiped.