Pakhrou, Tijani
Location problem and inner product spaces
Journal of Functional Analysis. 2023, 285(8): 110078. https://doi.org/10.1016/j.jfa.2023.110078
URI: http://hdl.handle.net/10045/135767
DOI: 10.1016/j.jfa.2023.110078
ISSN: 0022-1236 (Print)
Abstract: 
In this work we solve a problem that has been open for more than 110 years (see [21]). We prove that a real normed space (X, || · ||) of dimension greater than or equal to three is an inner product space if and only if, for every three points a1, a2, a3 ∈ X, the set of points at which the function x ∈ X → γ(||x − a1||, ||x − a2||, ||x − a3||) attains its minimum, intersects the convex hull of these three points, where γ is a symmetric monotone norm on R3.
Keywords:Optimal location, Chebyshev centers, Medians, Inner product spaces
Elsevier
info:eu-repo/semantics/article