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 ${\text{dist}}^2(x,y)$ is DC (d.c., delta-convex) on $X\times X$ in the only natural extrinsic sense. An analogous result holds for the squared distance function ${\text{dist}}^2(x,F)$ from a closed set $F\subset X$. Applications concerning $r$-boundaries (distance spheres) and the ambiguous locus (exoskeleton) of a closed subset of a convex surface are given.