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

A Minkowski space Md=(Rd, ‖ ‖) is just Rd with distances measured using a norm ‖ ‖. A norm ‖ ‖ is completely determined by its unit ball {x∈Rd∣‖x‖⩽1} which is a centrally symmetric convex body of the d-dimensional Euclidean space Ed. In this note we give upper bounds for the maximum number of times the minimum distance can occur among n points in Md, 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 Ed, d⩾3.