Functionally closed sets and functionally convex sets in real Banach spaces

Let x be a real normed space, then c(⊆ x) is functionally convex (briefly, f-convex), if t(c) ⊆ ris convex for all bounded linear transformations t ∈ b(x, r); and k(⊆ x) is functionally closed(briefly, f-closed), if t(k) ⊆ r is closed for all bounded linear transformations t ∈ b(x, r). weimprove the krein-milman theorem on finite dimensional spaces. we partially prove the chebyshev60 years old open problem. finally, we introduce the notion of functionally convex functions. thefunction f on x is functionally convex (briefly, 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.