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

Abstract

Let $F\subset {\mathbb{R}}^n$ 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 ${\mathbb{R}}^n\setminus F$. 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 ${\ell }_n^p,n=2,3,p\ge 2$), 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.

21 pages

References in corpus (2)

• Hamilton-Jacobi Equations and Distance Functions on Riemannian Manifolds
• Properties of distance functions on convex surfaces and applications

Cited by in corpus (3)

• Dimensions, Whitney covers, and tubular neighborhoods
• Normal cycles and curvature measures of sets with d.c. boundary
• Smallness of the set of critical values of distance functions in two-dimensional Euclidean and Riemannian spaces