POINTS JOINED BY 3 SHORTEST PATHS ON CONVEX SURFACES

Let S be a convex surface and x is an element of S. It is shown here t hat the set of all points of S joined with x by at least three shortes t paths can be dense in S. It is proven that, in fact, in the sense of Baire categories most convex surfaces have this property, for any x. Moreover, on most convex surfaces, for most of their points, there is just one farthest point (in the intrinsic metric), and precisely three shortest paths lead to that point.