Abstract
Let F be a closed subset of R^n 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 is an element of 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\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 is an element of {2, 3} (e.g., to l(n)(p), n = 2, 3, p }= 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.