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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 Bezdek--Pach Conjecture, homothets, packing, Hadwiger number, antipodality
MSC Classifications: 52C17, 51N20, 51K05, 52A21, 52A37 show english descriptions Packing and covering in $n$ dimensions [See also 05B40, 11H31]
Euclidean analytic geometry
General theory
Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx]
Other problems of combinatorial convexity
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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