Characterizations of inner product spaces by means of norm one points

Abstract

Let $X$ be a a real normed linear space of dimension at least three, with unit sphere $S_X$. In this paper we prove that $X$ is an inner product space if and only if every three point subset of $S_X$ has a Chebyshev center in its convex hull. We also give other characterizations expressed in terms of centers of three point subsets of $S_X$ only. We use in these characterizations Chebyshev centers as well as Fermat centers and $p$-centers.