On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body

Abstract

A Minkowski space M^d=(R^d, ‖ ‖) is just R^d with distances measured using a norm ‖ ‖. A norm ‖ ‖ is completely determined by its unit ball {x∈R^d∣‖x‖⩽1} which is a centrally symmetric convex body of the d-dimensional Euclidean space E^d. In this note we give upper bounds for the maximum number of times the minimum distance can occur among n points in M^d, d⩾3. In fact, we deal with a somewhat more general problem namely, we give upper bounds for the maximum number of touching pairs in a packing of n translates of a given convex body in E^d, d⩾3.