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On the Bezdek–Pach Conjecture for Centrally Symmetric Convex Bodies
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- Journal:
- Canadian Mathematical Bulletin / Volume 52 / Issue 3 / 01 September 2009
- Published online by Cambridge University Press:
- 20 November 2018, pp. 407-415
- Print publication:
- 01 September 2009
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- Article
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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.
