Question: Given a smooth, convex, closed surface. How would one find points on it with the property: the average of the geodesic distances from that point to all other points on the surface (the distance is of course, measured along the surface) is to be minimum/maximum?
Guesses: The point which minimizes the average geodesic distance from itself to all other points on the surface is one of the local minima of the Gaussian curvature; likewise, a local maximum of the Gaussian curvature will be the point that maximizes the average Gaussian distance.
Note: not sure if the global minimum and maximum of Gaussian curvature necessarily give the desired points. If that is indeed the case, if the closed and convex surface is polyhedral, the maximizing point would be that vertex where the sum of angles at that vertex of the faces meeting there would be the least. That the global maximum of curvature gives max average distance looks doubtful - in view of the fact that for a polygonal region, the longest diagonal need not begin from the vertex of the region with least angle.
Further query: Like the 4-vertex theorem, are there theorems on the minimum numbers of local extrema of Gaussian curvature on a 2D surface?
