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# On the Bezdek--Pach Conjecture for Centrally Symmetric Convex Bodies

Published:2009-09-01
Printed: Sep 2009
• Zsolt Lángi
• Márton Naszódi
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## Abstract

The Bezdek--Pach conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in $\Re^d$ is $2^d$. Nasz\'odi 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 $3\cdot2^{d-1}$ 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: Bezdek--Pach Conjecture, homothets, packing, Hadwiger number, antipodality
 MSC Classifications: 52C17 - Packing and covering in $n$ dimensions [See also 05B40, 11H31] 51N20 - Euclidean analytic geometry 51K05 - General theory 52A21 - Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 52A37 - Other problems of combinatorial convexity

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