Distance Function Associated to a Prox-regular set

In this paper, we provide in a general Hilbert space new characterizations of uniform prox-regularity involving outside but sufficiently close points of considered sets. We show that the complement of a prox-regular set is nothing but the union of closed balls with common radius. We derive from this that the prox-regularity of a given closed set is equivalent to the semiconvexity property of its distance function. Various estimates involving the metric projection mapping to a prox-regular set are also established.