On p-Convex, Proximally Smooth, Quasi-Convex, Strictly Quasi-Convex and Approximately Convex Sets

We compare four generalizations of convex sets which ensure good properties of the projection: p-convexity (widely identical to ρ-prox regularity), proximal smoothness, quasi-convexity and strict quasi-convexity, and one generalization based on the distance function: approximate convexity. We prove that (i) p-convex sets essentially coincide with quasi-convex sets, (ii) strictly quasi-convex sets are a subclass of proximally smooth sets and (iii) p-convex or quasi-convex sets are approximately convex, but the converse is false. The definition and main properties of these approaches are recalled without demonstration but with unified notations. We compare the Lipschitz properties of the projection on the different families of sets, and show that strict quasi-convexity ensures moreover the unimodality of the distance to a point over the set, and hence the computability of the projection by local optimization algorithms. Sufficient size-times-curvature conditions for strict quasi-convexity are also recalled.