On geometric properties of sets of positive reach in E d

Some geometric facts concerning sets with positive reach in Ed are proved. For x1 and x2 in E d and R > 0 let us denote by H(x1, x2, R) the intersection of all closed balls of radius R containing x1 and x2. We prove that reach(K) ≥ R if and only if for every x1, x2 ∈ K such that ‖x1 − x2‖ < 2R, H(x1, x2, R) ∩ K is connected. A corollary is that if reach(K) ≥ R > 0 and D is a closed ball of radius less than or equal to R (intersecting K) then reach(K ∩ D) ≥ R. For A ⊂ Ed and R > 0 we say that A admits R-hull if there extsts  ⊃ A, with reach(Â) ≥ R and such that  is the minimal set (with respect to inclusion) having these properties. A necessary and sufficient condition for a set A ⊂ Ed to admit a R-hull is provided. AMS 2000 Subject Classification: 52A30.