Points joined by three shortest paths on convex surfaces

Abstract:

Let S be a convex surface and $x \in S$. It is shown here that the set of all points of S joined with x by at least three shortest 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.