A note on the metric geometry of the unit ball

Abstract

Denote by Bn the unit ball in the Euclidean space \({\mathbb{R}^n}\) and define $$ M(B^n) = \sup \int_{B^n} \int_{B^n}\| x - y \| \, d\mu(x)d\mu(y),$$ where the supremum is taken over all finite signed Borel measures μ on Bn of total mass 1. In this paper, the value of M(Bn) is computed explicitly for all n, and it is shown that for n > 1 no measure exists that achieves the supremum defining M(Bn). These results generalize the work of Alexander (Proc Am Math Soc 64:317–320, 1977) on M(B3).