Tubular neighborhoods in Euclidean spaces (Q1075598)

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 \(\leq\).