Properties of distance functions on convex surfaces and applications

TL;DR

This paper investigates the delta-convexity of squared intrinsic distance functions on convex surfaces and explores applications to distance spheres and exoskeletons of closed subsets.

Contribution

It proves that squared intrinsic distances are delta-convex on convex surfaces and extends this to distances from closed sets, with applications to geometric structures.

Findings

Squared intrinsic distance functions are delta-convex on convex surfaces.

Results apply to distance spheres and exoskeletons of closed subsets.

Provides new insights into geometric properties of convex surfaces.

Abstract

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.