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.