Properties of distance functions on convex surfaces and applications

If X is a convex surface in a Euclidean space, then the squared intrinsic distance function dist²(x, y) is DC (d.c., delta-convex) on X×X in the only natural extrinsic sense. An analogous result holds for the squared distance function dist²(x,F) from a closed set F ⊂ X. Applications concerning r-boundaries (distance spheres) and ambiguous loci (exoskeletons) of closed subsets of a convex surface are given.