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

2012/02/02 by Sebastian Scholtes, Scholtes, Sebastian
Computer Science · Engineering · Mathematics · #46B20 #46C15 #Advanced Banach Space Theory #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematics and Applications #Metric Geometry (math.MG) #Optimization and Variational Analysis #Point processes and geometric inequalities #math.CA #math.FA #math.MG #msc:46B20 #msc:46C15

paper · pdf · doi:10.48550/arxiv.1202.0503

8 pages

arxiv created 2012/02/02 · openalex publication_date 2012/02/02 · arxiv updated 2012/02/03 · openalex created_date 2022/09/26 · openalex updated_date 2026/07/28

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⋅) with dim X≥ 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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