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

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

2009-11-20

arxiv.org/abs/0911.4020v1

Mathematics

Metric Geometry

21 pages

Scientific paper

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.

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