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Transactions of the American Mathematical Society

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Location of the Fermat-Torricelli medians of three points


Authors: Carlos Benítez, Manuel Fernández and María L. Soriano
Journal: Trans. Amer. Math. Soc. 354 (2002), 5027-5038
MSC (2000): Primary 46B20, 46C15, 90B85
Published electronically: August 1, 2002
MathSciNet review: 1926847
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Abstract: We prove that a real normed space $X$ with $\dim X\ge 3$ is an inner product space if and only if, for every three points $u,v,w\in X$, the set of points at which the function $x\in X\to \Vert u-x\Vert+\Vert v-x\Vert+\Vert w-x\Vert$attains its minimum (called the set of Fermat-Torricelli medians of the three points) intersects the convex hull of these three points.


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Additional Information

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

Manuel Fernández
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain
Email: ghierro@unex.es

María L. Soriano
Affiliation: Departamento de Matemáticas, Universidad de Extremadura, 06071 Badajoz, Spain
Email: lsoriano@unex.es

DOI: https://doi.org/10.1090/S0002-9947-02-03113-6
Keywords: Optimal location, medians, inner product spaces
Received by editor(s): November 27, 2000
Received by editor(s) in revised form: May 17, 2001
Published electronically: August 1, 2002
Additional Notes: Partially supported by MCYT (Spain) and FEDER, BFM2001-0849
Article copyright: © Copyright 2002 American Mathematical Society