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Aequationes mathematicae

, Volume 92, Issue 3, pp 529–541 | Cite as

Inscribed and circumscribed polygons that characterize inner product spaces

  • Carlos Benítez
  • Pedro Martín
  • Diego Yáñez
Article

Abstract

Let X be a real normed space with unit sphere S. We prove that X is an inner product space if and only if there exists a real number \(\rho =\sqrt{(1+\cos \frac{2k\pi }{2m+1})/2}, (k=1,2,\ldots ,m; m=1,2,\ldots )\), such that every chord of S that supports \(\rho S\) touches \(\rho S\) at its middle point. If this condition holds, then every point \(u\in S\) is a vertex of a regular polygon that is inscribed in S and circumscribed about \(\rho S\).

Keywords

Characterization Normed spaces Inner product 

Mathematics Subject Classification

46B20 46C15 52A10 52A21 

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad de ExtremaduraBadajozSpain

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