The Fermat--Torricelli problem under isosceles orthogonality-type conditions

Abstract

The location problem studied by Weber \cite{Weber} leads to the Fermat--Torricelli problem of minimizing sums of distances. Benítez, Fernández and Soriano \cite{BFS} proved that a real normed space of dimension at least three is an inner-product space if and only if every triple of points has a Fermat--Torricelli median in its convex hull. Pakhrou \cite{Pakhrou} showed that it suffices to test equilateral triangles. We establish a characterization using triangles $\{0,x,y\}$ satisfying \[ \|x+y\|=λ\|x-y\|,\qquad \forall\,λ>2/3, \] with the parameter fixed in each characterization. The case of parameter one gives an isosceles orthogonality criterion. The proof combines radial rescaling with the transfer of planar minimality to the ambient space, without completeness or regularity assumptions on the norm.