Flavors of Translative Coverings

TL;DR

This survey explores various methods for covering Euclidean space and convex bodies with translates, highlighting key results like Rogers' theorem and discussing related conjectures and problems in the field.

Contribution

It presents a diverse set of approaches to covering problems, emphasizing Rogers' theorem and related conjectures, offering a comprehensive overview of current methods.

Findings

Rogers' theorem states space can be covered with density around n log n.

Four different approaches to Rogers' theorem are outlined.

Discussion includes the illumination conjecture and covering problems.

Abstract

We survey results on the problem of covering the space ${\mathbb R}^n$, or a convex body in it, by translates of a convex body. Our main goal is to present a diverse set of methods. A theorem of Rogers is a central result, according to which, for any convex body $K$, the space ${\mathbb R}^n$ can be covered by translates of $K$ with density around $n\ln n$. We outline four approaches to proving this result. Then, we discuss the illumination conjecture, decomposability of multiple coverings, Sudakov's inequality and some problems concerning coverings by sequences of sets.