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

http://dx.doi.org/10.4153/CMB-2009-044-8
Canad. Math. Bull. 52(2009), 407-415

  Published:2009-09-01
 Printed: Sep 2009

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.