Some Properties in Fuzzy Convex Sets

In the basic and classical paper [10], where the important concept of fuzzy set was first introduced, L. A. Zadeh developed a basic framework to treat mathematically the fuzzy phenomena or systems which, due to intrinsic indefiniteness, cannot themselves be characterized precisely. He pays special attention to the investigation on the fuzzy convex sets which consists of nearly the second half of the space of his paper. The main results on fuzzy convex sets given in [10] are summarized as follows: (1) The separation theorem; and (2) The theorem on the shadows of fuzzy convex sets. The revised correct version of the separation theorem has been given in [9] by employing induced fuzzy topology. Using the concept of fuzzy hyperplane, Lowen has established some further separation theorem for fuzzy convex sets [6]. Concerning the theorem of shadow of fuzzy convex sets, Zadeh has further investigated in [11]. But in this respect, there exists still some drawbacks which will be shown via a counterexample in the present paper. Perhaps the lack of fuzzy topological assumption in the above mentioned results leads to the appearance of these shortcomings. Such a situation seems to be only natural in the early stage of development of the fuzzy set theory. Adding some assumptions about fuzzy topology, we are able to yield several positive results on the shadows of fuzzy convex sets. Finally we shall give some simple and direct proofs of two theorems that describe the relationships between the fuzzy convex cones and the fuzzy linear subspaces and that originally appeared in [6]. The present proofs do not appeal to the representation theorem established in [6] by R. Lowen.