Location of the Fermat-Torricelli medians of three points
Carlos Benítez, Manuel Fernández, M. Laura Soriano · Transactions of the American Mathematical Society · 2002
We prove that a real normed space X X with dim X ≥ 3 \dim X\ge 3 is an inner product space if and only if, for every three points u , v , w ∈ X u,v,w\in X , the set of points at which the function x ∈ X → ‖ u − x ‖ + ‖ v − x ‖ + ‖ w − x ‖ x\in X\to \|u-x\|+\|v-x\|+\|w-x\| attains its minimum (called the set of Fermat-Torricelli medians of the three points) intersects the convex hull of these three points.