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

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

2012-02-02

arxiv.org/abs/1202.0503v1

Mathematics

Functional Analysis

8 pages

Scientific paper

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.

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