The θθ-exterior sphere condition, φφ-convexity, and local semiconcavity

We consider sets S⊂RnS⊂Rn satisfying a locally uniform exterior sphere condition called the θθ-exterior sphere condition  . It is shown that under wedgedness of SS, this coincides with the φφ-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 SS, as well as semiconcavity and other properties of functions with hypographs satisfying the θθ-exterior sphere condition.