Flavors of Translative Coverings

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.