Well-Posedness of Approximation and Optimization Problems for Weakly Convex Sets and Functions

We consider the class of weakly convex sets with respect to a quasiball in a Banach space. This class generalizes the classes of sets with positive reach, proximal smooth sets and prox-regular sets. We prove the well-posedness of the closest points problem of two sets, one of which is weakly convex with respect to a quasiball M , and the other one is a summand of the quasiball −rM , where r ∈ (0 , 1). We show that if a quasiball B is a summand of a quasiball M , then a set that is weakly convex with respect to the quasiball M is also weakly convex with respect to the quasiball B . We consider the class of weakly convex functions with respect to a given convex continuous function γ that consists of functions whose epigraphs are weakly convex sets with respect to the epigraph of γ . We obtain a sufficient condition for the well-posedness of the infimal convolution problem, and also a sufficient condition for the existence, uniqueness, and continuous dependence on parameters of the minimizer.