On scalar metrics that maximize geodesic distances in the plane

A Riemannian metric a in the plane together with a point $${A\subset \mathbb {R}^2}$$ induces a distance function d a (A, ·). We investigate the optimization problem searching a scalar metric a which maximizes the distance between A and a given set B. We find necessary conditions for optimal metrics which help to determine solutions a. In the case that the set B is a single point, we determine the optimal metric explicitly.