Coverings: variations on a result of Rogers and on the Epsilon-net theorem of Haussler and Welzl

TL;DR

This paper explores advanced covering problems in geometry, extending classical results by Rogers and Haussler-Welzl, including space coverings, multiple coverings, and sphere coverings, with new bounds and combinatorial proofs.

Contribution

It introduces novel extensions of Rogers' and Haussler-Welzl's theorems, providing improved bounds and new methods for covering convex bodies and spheres.

Findings

Extended Rogers' result to infinite volume homothets

Provided bounds for multiple coverings of space

Demonstrated sphere covering with strips of specific width

Abstract

We consider four problems. Rogers proved that for any convex body $K$, we can cover ${\mathbb R}^d$ by translates of $K$ of density very roughly $d\ln d$. First, we extend this result by showing that, if we are given a family of positive homothets of $K$ of infinite total volume, then we can find appropriate translation vectors for each given homothet to cover ${\mathbb R}^d$ with the same (or, in certain cases, smaller) density. Second, we extend Rogers' result to multiple coverings of space by translates of a convex body: we give a non-trivial upper bound on the density of the most economical covering where each point is covered by at least a certain number of translates. Third, we show that for any sufficiently large $n$, the sphere ${\mathbb S}^2$ can be covered by $n$ strips of width $20n/\ln n$, where no point is covered too many times. Finally, we give another proof of the previous result based on a combinatorial observation: an extension of the Epsilon-net Theorem of Haussler and Welzl. We show that for a hypergraph of bounded Vapnik--Chervonenkis dimension, in which each edge is of a certain measure, there is a not-too large transversal set which does not intersect any edge too many times.