A remark on characterizing inner product spaces via strong three-point homogeneity

Abstract

We show that a normed linear space is isometrically isomorphic to an inner product space if and only if it is a strongly $n$-point homogeneous metric space for any (or every) $n⩾3$. The counterpart for $n=2$ is the Banach-Mazur problem.

Significantly expanded, including adding details in the proof of Theorem 4; the statement of Theorem 10; and all of Section 3. Final version, 9 pages, to appear in the Canadian Mathematical Bulletin