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

Sebastian Scholtes

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arXiv · 1202.0503

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

Sebastian Scholtes

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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.

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Sebastian Scholtes. 2012-02-02. A characterisation of inner product spaces by the maximal circumradius of spheres. https://arxiv.org/abs/1202.0503

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