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Middle points, medians and inner products


Authors: Carlos Benítez and Diego Yáñez
Journal: Proc. Amer. Math. Soc. 135 (2007), 1725-1734
MSC (2000): Primary 49B20, 46C15, 90B85
DOI: https://doi.org/10.1090/S0002-9939-06-08647-3
Published electronically: November 14, 2006
MathSciNet review: 2286082
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Abstract: Let $ X$ be a real normed space with unit sphere $ S$. Gurari and Sozonov proved that $ X$ is an inner product space if and only if, for any $ u,v\in S$, $ \inf_{t\in[0,1]}\Vert tu+(1-t)v\Vert= \Vert\tfrac12u+\tfrac12v\Vert$. We prove that it suffices to consider points $ u,v\in S$ such that $ \inf_{t\in[0,1]}\Vert tu+(1-t)v\Vert=\tfrac12$.

Making use of the above result we also prove that if $ \dim X\geq 3$, $ X$ is smooth, and 0 is a Fermat-Torricelli median of any three points $ u,v,w\in S$ such that $ u+v+w=0$, then $ X$ is an inner product space.


References [Enhancements On Off] (What's this?)

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Additional Information

Carlos Benítez
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain
Email: cabero@unex.es

Diego Yáñez
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain
Email: dyanez@unex.es

DOI: https://doi.org/10.1090/S0002-9939-06-08647-3
Received by editor(s): July 13, 2005
Received by editor(s) in revised form: December 26, 2005
Published electronically: November 14, 2006
Additional Notes: This work was partially supported by MEC (Spain) and FEDER (UE), MTM2004-06226
Communicated by: N. Tomczak-Jaegermann
Article copyright: © Copyright 2006 American Mathematical Society

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