Critical values and level sets of distance functions in Riemannian, Alexandrov and Minkowski spaces

TL;DR

This paper demonstrates that the topological structure of distance level sets in various geometric spaces can be understood through the properties of the distance function, extending classical results to more general settings like Riemannian and Alexandrov spaces.

Contribution

The authors show that Ferry's classical results follow from the local DC property of the distance function and extend these results to smooth normed spaces, Riemannian manifolds, and Alexandrov spaces.

Findings

Level sets are topological manifolds for almost all radii in specified spaces.

Extension of classical results to smooth normed spaces of dimensions 2 and 3.

Generalization of results to Riemannian and Alexandrov spaces.

Abstract

Let $F \subset \R^n$ be a closed set and $n=2$ or $n=3$. S. Ferry (1975) proved that then, for almost all $r>0$, the level set (distance sphere, $r$-boundary) $S_r(F):= \{x \in \R^n: \dist(x,F) = r\}$ is a topological $(n-1)$-dimensional manifold. This result was improved by J.H.G. Fu (1985). We show that Ferry's result is an easy consequence of the only fact that the distance function $d(x)= \dist(x,F)$ is locally DC and has no stationary point in $\R^n\setminus F$. Using this observation, we show that Ferry's (and even Fu's) result extends to sufficiently smooth normed linear spaces $X$ with $\dim X \in \{2,3\}$ (e.g., to $\ell^p_n, n=2,3, p\geq 2$), which improves and generalizes a result of R. Gariepy and W.D. Pepe (1972). By the same method we also generalize Fu's result to Riemannian manifolds and improve a result of K. Shiohama and M. Tanaka (1996) on distance spheres in Alexandrov spaces.