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

Some geometric facts concerning sets with positive reach in the Euclidean $d$-dimensional space ${\bf E}^d$ are proved. For $x_1$ and $x_2$ in ${\bf E}^d$ 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 ${bf E}^d$ we prove that ${\rm reach}(K)\ge R$ 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 ${\rm reach}(K)\ge R>0$ and $D$ is a closed ball of radius less than or equal to $R$ (intersecting $K$) then ${\rm reach}(K\cap D)\ge R$. We also give a necessary and sufficient condition such that $A\subset{\bf E}^d$ admits a minimal cover (with respect to inclusion) of ${\rm reach}\ge R$.