SCHWEITZER 2013 Problem 10

Consider a Riemannian metric on the vector space ${\Bbb{R}^n}$ which satisfies the property that for each two points ${a,b}$ there is a single distance minimising geodesic segment ${g(a,b)}$ . Suppose that for all ${a \in \Bbb{R}^n}$ , the Riemannian distance with respect to ${a}, {\rho_a : \Bbb{R}^n \rightarrow \Bbb{R}}$ is convex and differentiable outside of ${a}$ . Prove that if for a point ${x \neq a,b}$ we have $$\partial_i \rho_a(x)=-\partial_i \rho_b(x),\ i=1,\cdots, n$$ then ${x}$ is a point on ${g(a,b)}$ and conversely.

Proposed by Lajos Tamássy and Dávid Kertész