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 $\dist^2(x,y)$ 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 $\dist^2(x,F)$ 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.

💡 Research Summary

The paper investigates the regularity properties of intrinsic distance functions on convex surfaces embedded in Euclidean space and derives several geometric applications. The authors work with an n‑dimensional closed convex surface (X\subset\mathbb R^{n+1}) and a non‑empty closed subset (F\subset X). Their main achievement is to prove that the squared intrinsic distance \