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 $r\in(0,1)$. 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 $\gamma$ that consists of functions whose epigraphs are weakly convex sets with respect to the epigraph of $\gamma$. 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.