The translative kissing number of the Cartesian product of two convex bodies, one of which is two-dimensional (Q1357625)

For a given convex body \(K\) in \(\mathbb{R}^n\), the translative kissing number \(N(K)\) of \(K\) is defined as the maximum number of non-overlapping translates of \(K\) which can touch \(K\). As is well known, in \(\mathbb{R}^2\) \(N(K)=8\) if \(K\) is a parallelogram and \(N(K)=6\) otherwise. For arbitrary dimension, the exact value of \(N(K)\) is known only when \(K\) is the unit ball and the dimension of the space is equal to 3, 8 or 24. The author studies the translative kissing number of the Cartesian product of two convex bodies \(K_1\), \(K_2\) in \(\mathbb{R}^m\) and \(\mathbb{R}^n\) respectively. In particular he proves that if \(m\leq 2\) or \(n\leq 2\), then \[ N(K_1\oplus K_2)= \bigl(N(K_1) +1\bigr) \bigl(N(K_2) +1\bigr) -1. \] As an application of this formula, he finds the translative kissing number of \(B\oplus Q\), where \(B\) is a 24-dimensional ball and \(Q\) is a two-dimensional convex body. For arbitrary dimensions \(m\) and \(n\), the author provides an upper bound for \(N(K_1 \oplus K_2)\) which involves the cardinality of ``suitable'' partitions of one of the two bodies \(K_1\), \(K_2\). He also conjectures that for some large integers \(m\) and \(n\), there exist an \(m\)-dimensional body \(K_1\) and an \(n\)-dimensional body \(K_2\) such that \(N(K_1\oplus K_2) \neq(N(K_1) +1)(N(K_2) +1)-1\).