Functionally closed sets and functionally convex sets in real Banach spaces

Let X be a real normed space, then C(subset of X) is functionally convex (briefly, F-convex), if T(C) subset of R is convex for all bounded linear transformations T is an element of B(X, R); and K(subset of X) is functionally closed (briefly, F-closed), if T(K) subset of R is closed for all bounded linear transformations T is an element of B(X, R). We improve the Krein-Milman theorem on finite 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 (briefly, F-convex) if epi f is a F-convex subset of X x R. We show that every function f : (a, b) -> R which has no vertical asymptote is F-convex.