arXiv 1603.044812016-03-14math.MG

Flavors of Translative Coverings

Márton Naszódi

备注：: A survey, 24 pages

英文摘要：

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.

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