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Tubular neighborhoods in Euclidean spaces. (English) Zbl 0592.52002
This paper represents parts of the author’s dissertation and comprises five paragraphs: Semiconcave functions and their basic properties; Semiconcave functions and sets of positive reach; Implicit function theorem; Critical values of the distance function; Additional remarks.
For $$\alpha >0$$, let $${\mathfrak H}^{\alpha}$$ denote Hausdorff $$\alpha$$- dimensional measure on $${\mathbb{R}}^ n$$. Let $$S\subset {\mathbb{R}}^ n$$ be compact. Let $$S_ r:=\{x\in {\mathbb{R}}^ n: dis\tan ce\quad (x,S)\leq r\}$$ be the tubular neighborhood of S.
The main result is the following Theorem: There is a compact set $$C=C(S)\subset [0,(n/2n+2)^{1/2} diam(S)]$$ with $${\mathfrak H}^{(n- 1)/2}(C)=0$$, such that if $$r\not\in C$$ then the boundary of $$S_ r$$ is a Lipschitz manifold and $$\overline{({\mathbb{R}}^ n\setminus S_ r)}$$ is a set of positive reach. If $$n=2$$ then for every $$\epsilon >0$$ the entropy dimension of $$C\setminus [0,\epsilon)$$ is $$\leq$$.
Reviewer: C.Udrişte

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 26B25 Convexity of real functions of several variables, generalizations 28A75 Length, area, volume, other geometric measure theory 49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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