The distance function in the presence of an obstacle

TL;DR

This paper analyzes the regularity and singularity propagation of the Riemannian distance function in Euclidean space with obstacles, revealing optimal regularity near obstacles and the behavior of singularities.

Contribution

It establishes the local semiconcavity of the distance function with fractional modulus, characterizes singularity propagation, and examines differentiability issues near convex obstacles.

Findings

Distance function is locally semiconcave with fractional modulus.

Singularities propagate along nontrivial continua.

Distance function may lack differentiability near convex obstacles.

Abstract

We study the Riemannian distance function from a fixed point (a point-wise target) of Euclidean space in the presence of a compact obstacle bounded by a smooth hypersurface. First, we show that such a function is locally semiconcave with a fractional modulus of order one half and that, near the obstacle, this regularity is optimal. Then, in the Euclidean setting, we prove that the singularities of the distance function propagate, in the sense that each singular point belongs to a nontrivial singular continuum. Finally, we investigate the lack of differentiability of the distance function when a convex obstacle is present.