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 $u,v\in S$, $\inf _{t\in [0,1]}\|tu+(1-t)v\|= \|\tfrac 12u+\tfrac 12v\|$. We prove that it suffices to consider points $u,v\in S$ such that $\inf _{t\in [0,1]}\|tu+(1-t)v\|=\tfrac 12$. Making use of the above result we also prove that if $\dim X\geq 3$, $X$ is smooth, and 0 is a Fermat-Torricelli median of any three points $u,v,w\in S$ such that $u+v+w=0$, then $X$ is an inner product space.