On geometric properties of sets of positive reach in ${\mathbf{E}}^d$

Abstract

Some geometric facts concerning sets with positive reach in the Euclidean $d$-dimensional space ${\mathbf{E}}^d$ are proved. For $x_1$ and $x_2$ in ${\mathbf{E}}^d$ and $R>0$ let us denote by $\mathfrak{H}(x_1,x_2,R)$ the intersection of all closed balls of radius $R$ containing $x_1$ and $x_2$. For a compact subset $K$ of ${bfE}^d$ we prove that $\mathrm{reach}(K)\ge R$ if and only if for every $x_1,x_2\in K$ such that $\Vert x_1-x_2\Vert <2R$, $\mathfrak{H}(x_1,x_2,R)\cap K$ is connected. A corollary is that if $\mathrm{reach}(K)\ge R>0$ and $D$ is a closed ball of radius less than or equal to $R$ (intersecting $K$) then $\mathrm{reach}(K\cap D)\ge R$. We also give a necessary and sufficient condition such that $A\subset {\mathbf{E}}^d$ admits a minimal cover (with respect to inclusion) of $\mathrm{reach}\ge R$.

10 pages, 2 figures