A characterisation of inner product spaces by the maximal circumradius of spheres

TL;DR

This paper characterizes inner product spaces among normed vector spaces by showing that such spaces have non-degenerate spheres where the maximal circumradius equals the sphere's radius.

Contribution

It introduces a novel geometric criterion based on the maximal circumradius of spheres to identify inner product spaces.

Findings

Inner product spaces have non-degenerate spheres with maximal circumradius equal to the sphere radius.

The characterization applies to normed spaces with dimension at least 2.

The criterion provides a geometric test for inner product space structure.

Abstract

We give a new characterisation of inner product spaces amongst normed vector spaces in terms of the maximal cirumradius of spheres. It turns out that a normed vector space $(X,\norm{\cdot})$ with $\dim X\geq 2$ is an inner product space if and only if all spheres are not degenerate, i.e. the maximal circumradius of points on the sphere equals the radius of the sphere.