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

Abstract

The µ-kernel of an o-symmetric convex body is obtained by shrinking the body about its center by a factor of µ. As a generalization of pairwise intersecting Minkowski arrangement of o-symmetric convex bodies, we can define the pairwise intersecting Minkowski arrangement of order µ. Here, the homothetic copies of an o-symmetric convex body are so that none of their interiors intersect the µ-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 µ = 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 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.