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 <f>x,y</f> in the domain and for all <f>t</f> in <f>]0,1[ …</f>” by the corresponding conditions of type “for all <f>x,y</f> in the domain there exists <f>t</f> in <f>]0,1[ … </f>.” [Copyright &y& Elsevier]