On the geometry of sets of positive reach
Jennifer Carol Ellis · 2012
The reach of a set S in a metric space, denoted reach(S), is the supremal r such that any point within distance r of S has a unique nearest point in S. Sets of positive reach (PR sets) originate in the work of Federer in [9] with a theory of curvature measures on PR sets. More recently, Fu [10] defined the second fundamental form for PR sets, established Morse theory on PR sets, and revisited Federer’s curvature measures. We work exclusively with regular PR sets in Euclidean space. We further develop the theory of regular PR sets in Euclidean space by establishing regularity of geodesics and by determining a formula for reach using the second fundamental form. We prove that geodesics are C1,1 in regular PR sets. Our formula for reach of regular compact PR sets is based on the technique in [3] for determining thickness of C1,1 curves.