The separation of convex sets and the Krein–Milman theorem in fuzzy quasi-normed space

Motivated by some deep problems in optimization and control theory, convexity theory has been extended to the various infinite dimensional functional spaces. The separation theorems play key roles in developing convexity theory. In this paper, we focus on the convexity problem in the framework of fuzzy quasi-normed spaces. First, we give some separation results of convex sets, and show a characterization of a closed convex subset with the aid of the dual of a fuzzy quasi-normed space. Second, we introduce the concept of an extreme point and generalize the famous Krein–Milman theorem to a fuzzy quasi-normed space. The obtained results of this paper will play key roles in convex programming and optimization problems of fuzzy quasi-normed spaces.