The weighted Fermat–Torricelli problem on a surface and an “inverse” problem

We study the weighted Fermat–Torricelli (w.F-T) problem for geodesic triangles on a C2 complete surface and on an Aleksandrov space of curvature bounded above by a real number K and solve an “inverse” problem on a C2 complete surface. The solution of the w.F-T problem and the inverse w.F-T problem on a C2 complete surface is based on the differentiation of the length of geodesics with respect to the arc length.