The θ-exterior sphere condition, φ-convexity, and local semiconcavity

Abstract

We consider sets S⊂Rn satisfying a locally uniform exterior sphere condition called the θ-exterior sphere condition . It is shown that under wedgedness of S, this coincides with the φ-convexity property studied by Colombo and Marigonda (2005) [5]. This generalizes and improves upon the main result of Nour et al. (2009) [10], where the constant radius case was considered. We also generalize certain known results concerning the regularity of the distance function associated with S, as well as semiconcavity and other properties of functions with hypographs satisfying the θ-exterior sphere condition.