Flavors of Translative Coverings

  • Márton NaszódiEmail author
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 27)


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.


Covering Rogers’ bound Spherical cap Density Set-cover Illumination Borsuk’s conjecture Multiple covering Sudakov’s inequality 

2010 Mathematics Subject Classification

52C17 05B40 52A23 



The author is grateful for the many illuminating conversations with Gábor Fejes Tóth about covering problems in general, and about this manuscript.


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© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of GeometryLorand Eötvös UniversityBudapestHungary

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