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.