Pairwise intersecting homothets of a convex body (Q5915835)

A \textit{Minkowski arrangement} of a \(0\)-symmetric convex body \(K\) is a family \(\{v_i+\lambda_i K\}\) of homothets of \(K\), with \(\lambda_i>0\) and \(v_i\in{\mathbb R}\), such that none of them contains in its interior the center of any other set of the family. In the paper under review the author proves that the largest number of homothetic copies that a pairwise intersecting Minkowski arrangement of \(K\) can have is bounded from above by \(3^{d+1}\). The paper is also devoted to study the so-called \(k\)-distance sets in Minkowski spaces: a subset \(S\) of a metric space is a \textit{\(k\)-distance set}, \(k>1\), if the collection of non-zero distances occurring between points of \(S\) has size at most \(k\). The author shows that the cardinality of a \(k\)-distance set in a \(d\)-dimensional Minkowski space is at most \(k^{O(3^dd)}\), the constant in \(O(\cdot)\) being independent of \(d\) and \(k\).