
Journal of Convex Analysis 22 (2015), No. 2, 427446 Copyright Heldermann Verlag 2015 On pConvex, Proximally Smooth, QuasiConvex, Strictly QuasiConvex and Approximately Convex Sets Guy Chavent Ceremade, Université ParisDauphine, 75775 Paris Cedex 16, France guy.chavent@free.fr We compare four generalizations of convex sets which ensure good properties of the projection: pconvexity (widely identical to ρprox regularity), proximal smoothness, quasiconvexity and strict quasiconvexity, and one generalization based on the distance function: approximate convexity. We prove that (i) pconvex sets essentially coincide with quasiconvex sets, (ii) strictly quasiconvex sets are a subclass of proximally smooth sets and (iii) pconvex or quasiconvex 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 quasiconvexity 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 sizetimescurvature conditions for strict quasiconvexity are also recalled. Keywords: Generalized set convexity, pconvexity, proxregularity, approximate convexity, proximal smoothness, quasiconvexity, strict quasiconvexity, geodesics, singlevalued projection, closest points, Hilbert space. [ Fulltextpdf (192 KB)] for subscribers only. 