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On the Bezdek–Pach Conjecture for Centrally Symmetric Convex Bodies

Published online by Cambridge University Press:  20 November 2018

Zsolt Lángi  and
Márton Naszódi
Zsolt Lángi
Affiliation:
Department of Geometry, BudapestUniversity of Technology, Budapest, Egry József u. 1., Hungary, 1111 e-mail: zlangi@math.bme.hu
Márton Naszódi
Affiliation:
Department of Math. and Statistics, University of Alberta, Edmonton, AB, T6G 2G1 e-mail: mnaszodi@math.ualberta.ca
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Abstract

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The Bezdek–Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${{\mathbb{R}}^{d}}$ is ${{2}^{d}}$. Naszódi proved that the quantity in question is not larger than ${{2}^{d+1}}$. We present an improvement to this result by proving the upper bound $3\,\cdot \,{{2}^{d-1}}$ for centrally symmetric bodies. Bezdek and Brass introduced the one-sided Hadwiger number of a convex body. We extend this definition, prove an upper bound on the resulting quantity, and show a connection with the problem of touching homothetic bodies.

Keywords

52C1751N2051K0552A2152A37Bezdek–Pach conjecturehomothetspackingHadwiger numberantipodality
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 52 , Issue 3 , 01 September 2009 , pp. 407 - 415
Copyright
Copyright © Canadian Mathematical Society 2009

References

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