Through this study, we are going to introduce a new aspect of cut sets for hesitant fuzzy sets. Compared to the rare researches in regard of cut sets for hesitant fuzzy sets, not only this study does not deal with extra parameters, but also the proposed concept of cut set conforms with the accepted principles, and further, it is extension of existing cut set concept of fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets, interval-valued intuitionistic fuzzy sets and three-dimensional fuzzy sets. In the sequel, using proposed concept of cut sets for hesitant fuzzy sets, we extend the corresponding theorems of decomposition and representation for hesitant fuzzy sets. The current research can be a basis for the future work on the application of cut sets in multiple criteria decision making problems under hesitant fuzzy environment.

Let X be a real normed space, then C( X) is functionally convex (brie y, F-convex), if T(C)  R is convex for all bounded linear transformations T 2 B(X; R); and K( X) is functionally closed (brie y, F-closed), if T(K)  R is closed for all bounded linear transformations T 2 B(X; R). We improve the Krein-Milman theorem on  nite dimensional spaces. We partially prove the Chebyshev 60 years old open problem. Finally, we introduce the notion of functionally convex functions. The function f on X is functionally convex (brie y, F-convex) if epi f is a F-convex subset of X  R. We show that every function f: (a; b) 􀀀 ! R which has no vertical asymptote is F-convex.

In this paper, a new definition of fuzzy bounded sets and totally fuzzy bounded sets is introduced and properties of such sets are studied. Then a relation between totally fuzzy bounded sets and N-compactness is discussed. Finally, a geometric characterization for fuzzy totally bounded sets in I- topological vector spaces is derived.