Location problem and inner product spaces

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.