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1. CMB 2009 (vol 52 pp. 407)

Lángi, Zsolt; Naszódi, Márton
On the Bezdek--Pach Conjecture for Centrally Symmetric Convex Bodies
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
Categories:52C17, 51N20, 51K05, 52A21, 52A37

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