Convexity and semicontinuity of fuzzy sets

In this paper we study a modified concept called fuzzy convexity which was proposed by Ammar and Metz (Fuzzy Sets and Systems 49 (1992) 135). A criteria for convex fuzzy sets under lower semicontinuity is given. We prove in the upper semicontinuous case, that the class of semistrictly quasiconvex fuzzy sets lies between the quasiconvex and strictly quasiconvex classes. We also prove for both families of semistrictly quasiconvex and strictly quasiconvex fuzzy sets, that every local maximizer is also a global one. In addition, a characterization of quasimonotonic fuzzy sets in terms of level sets is given.