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 $Tin B(X,R)$; and $K(subseteq X)$  is  functionally   closed (briefly, $F$-closed), if  $T(K)subseteq Bbb R $ is  closed  for all bounded linear transformations $Tin B(X,R)$. 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 $Xtimes mathbb{R}$. We show that every  function $f : (a,b)longrightarrow mathbb{R}$ which has no  vertical asymptote is $F$-convex.