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On some covering problems in geometry


Author: Márton Naszódi
Journal: Proc. Amer. Math. Soc. 144 (2016), 3555-3562
MSC (2010): Primary 52C17, 05B40, 52A23
DOI: https://doi.org/10.1090/proc/12992
Published electronically: January 26, 2016
MathSciNet review: 3503722
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Abstract: We present a method to obtain upper bounds on covering numbers. As applications of this method, we reprove and generalize results of Rogers on economically covering Euclidean $ n$-space with translates of a convex body, or more generally, any measurable set. We obtain a bound for the density of covering the $ n$-sphere by rotated copies of a spherically convex set (or, any measurable set). Using the same method, we sharpen an estimate by Artstein-Avidan and Slomka on covering a bounded set by translates of another.

The main novelty of our method is that it is not probabilistic. The key idea, which makes our proofs rather simple and uniform through different settings, is an algorithmic result of Lovász and Stein.


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Additional Information

Márton Naszódi
Affiliation: Department of Geometry, Lorand Eötvös University, Pázmány Péter Sétány 1/C Budapest, Hungary 1117
Email: marton.naszodi@math.elte.hu

DOI: https://doi.org/10.1090/proc/12992
Keywords: Covering, Rogers' bound, spherical cap, density, set-cover
Received by editor(s): March 30, 2015
Received by editor(s) in revised form: October 1, 2015
Published electronically: January 26, 2016
Additional Notes: The author acknowledges the support of the János Bolyai Research Scholarship of the Hungarian Academy of Sciences, and the Hung. Nat. Sci. Found. (OTKA) grant PD104744.
Communicated by: Patricia L. Hersh
Article copyright: © Copyright 2016 American Mathematical Society