The Nearest Point Theorem for Weakly Convex Sets in Asymmetric Seminormed Spaces

Abstract

Weakly convex sets in asymmetric seminormed spaces are considered. We prove that any point from some neighborhood of such a set has the unique nearest point in the set. The proof of the nearest point theorem is based on the theorem about the diameter of \(\varepsilon \)-projection which is also important in approximation theory. The notion of weakly convex sets in asymmetric seminormed spaces generalizes known notions of sets with positive reach, proximal smooth sets, and prox-regular sets. By taking the Minkowski functional of the epigraph of some convex function as a seminorm, the results obtained for weakly convex sets can be applied to weakly convex functions whose graphs are weakly convex sets with respect to this seminorm.