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ARRANGEMENTS OF HOMOTHETS OF A CONVEX BODY

Part of: Normed linear spaces and Banach spaces; Banach lattices Discrete geometry General convexity

Published online by Cambridge University Press:  09 August 2017

Márton Naszódi ,
János Pach  and
Konrad Swanepoel
Márton Naszódi
Affiliation:
Department of Geometry, Lorand Eötvös University, Pazmány Péter Sétany 1/C Budapest, 1117, Hungary email marton.naszodi@math.elte.hu
János Pach
Affiliation:
École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland email pach@cims.nyu.edu Rényi Institute, Budapest, Hungary
Konrad Swanepoel
Affiliation:
Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, U.K. email k.swanepoel@lse.ac.uk
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Abstract

Answering a question of Füredi and Loeb [On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc.121(4) (1994), 1063–1073], we show that the maximum number of pairwise intersecting homothets of a $d$ -dimensional centrally symmetric convex body  $K$ , none of which contains the center of another in its interior, is at most $O(3^{d}d\log d)$ . If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^{d}\binom{2d}{d}d\log d)$ . We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$ . We also show that, in the latter case, one can always find families of at least $\unicode[STIX]{x1D6FA}((2/\sqrt{3})^{d})$ translates of $K$ with the above property.

MSC classification

Primary: 52C17: Packing and covering in $n$ dimensions
Secondary: 46B06: Asymptotic theory of Banach spaces 52A23: Asymptotic theory of convex bodies

Information

Type
Research Article
Information
Mathematika , Volume 63 , Issue 2 , 2017 , pp. 696 - 710
Copyright
Copyright © University College London 2017 

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