The distance function from the boundary of a domain with corners

TL;DR

This paper investigates the regularity and second derivatives of distance functions to domains with corners under asymmetric norms, and characterizes the ridge set using generalized curvature.

Contribution

It provides explicit formulas for second derivatives of distance functions and characterizes the ridge set in domains with corners, extending previous regularity results.

Findings

Explicit second derivative formulas for distance functions.

Complete characterization of the ridge set via generalized curvature.

Analysis of distance functions in domains with corners under asymmetric norms.

Abstract

We study the regularity of the distance function to the boundary of a domain in $\mathbb{R}^2$, with respect to some asymmetric norms. We allow the boundary of the domain to have corners. We obtain an explicit formula for the second derivative of these distance functions. Furthermore, we study a generalized notion of the ridge of a domain, which is the set of singularities of a distance function to the boundary of the domain. We completely characterize the ridge by using a generalized notion of curvature.