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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

\displaystyle \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

Problem Statement