Functionally closed sets and functionally convex sets in real Banach spaces

‎Let X be a real normed  space, then  C(\subseteq X)  is  functionally  convex  (briefly, F-convex), if  T(C)\subseteq \Bbb R is  convex for all bounded linear transformations T\in B(X,R); and K(\subseteq X)  is  functionally   closed (briefly, F-closed), if  T(K)\subseteq \Bbb R is  closed  for all bounded linear transformations T\in B(X,R). We improve the    Krein-Milman theorem  on finite dimensional spaces. We partially prove the Chebyshev ۶۰ 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\times \mathbb{R}. We show that every  function f : (a,b)\longrightarrow \mathbb{R} which has no  vertical asymptote is F-convex.