Bounds on Convex Bodies in Pairwise Intersecting Minkowski Arrangement of Order $\mu$

Abstract

A generalization of pairwise intersecting Minkowski arrangement of centrally symmetric convex bodies is the pairwise intersecting Minkowski arrangement of order $\mu$. Here, the homothetic copies of a centrally symmetric convex body are so that none of their interiors intersect the $\mu$-kernel of any other. We give general upper and lower bounds on the cardinality of such arrangements, and study two special cases: For $d$-dimensional translates in classical pairwise intersecting Minkowski arrangement we prove that the sharp upper bound is $3^d$. For $\mu =1$ the general version yields to another known problem: The Bezdek-Pach Conjecture asserts that the maximum number of pairwise touching positive homothetic copies of a convex body in ${\mathbb{R}}^d$ is $2^d$. We verify the conjecture on the plane, that is, when $d=2$. Indeed, we show that the number in question is four for any planar convex body.