Points joined by three shortest paths on convex surfaces
Tudor I. Zamfirescu · Proceedings of the American Mathematical Society · 1995
Let S be a convex surface and x ∈ S 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.