Generalized convex set-valued maps ✩

The aim of this paper is to show that under a mild semicontinuity assumption (the so-called segmentary epi-closedness), the cone-convex (respectively, cone-quasiconvex) set-valued maps can be characterized in terms of weak cone-convexity (respectively, weak cone-quasiconvexity), i.e., the notions obtained by replacing in the classical definitions the conditions of type “for all x, y in the domain and for all t in ]0, 1[ . . .” by the corresponding conditions of type “for all x, y in the domain there exists t in ]0, 1[ . . . .”  2003 Elsevier Inc. All rights reserved.