On a conjecture of Károly Bezdek and János Pach

Summary

The following conjecture of K\'aroly Bezdek and J\'anos Pach is cited in~[1]. If <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"3"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"4"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"5"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"6"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>K\subset{\mathbb R}^d$ is a convex body then any packing of pairwise touching positive homothets of $K$ consists of at most $2^d$ copies of $K$. We prove a weaker bound, $2^{d+1}$.