Abstract
We consider sets S ⊂ R n 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.
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