Middle points, medians and inner products

Abstract

Let X be a real normed space with unit sphere S. Gurari and Sozonov proved that X is an inner product space if and only if, for any u, v ∈ S, inf t∈ [0,1] ∥tu + (1 - t)v∥ = ∥½u + ½v∥. We prove that it suffices to consider points u,v ∈ S such that inf t∈ [0,1] ∥tu+ (1 - t)v∥ = ½. Making use of the above result we also prove that if dim X > 3, X is smooth, and 0 is a Fermat-Torricelli median of any three points u, v, w ∈ S such that u + v + w = 0, then X is an inner product space.