Flavors of Translative Coverings

2016/03/14 by Márton Naszódi, Naszódi, Márton
Mathematics · #05B40 #52A23 #52C17 #FOS: Mathematics #Metric Geometry (math.MG) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1603.04481

openalex publication_date 2016/03/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

We survey results on the problem of covering the space \mathbb Rⁿ, 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 Rⁿ can be covered by translates of K with density around nln 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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