Given a convex function $f : A \to \Bbb{R}$, where $A$ is a convex compact set from $\Bbb{R}^n$. How to prove that we can extend our function $F : \Bbb{R}^n \to \Bbb{R}$, such that it stays convex.
We think that the solution is F(x) = sup{λf(y) + (1 − λ)f(z): x = λy + (1 − λ)z, y, z ∈ Ω, λ >= 1}, it is clear that F(x) extends f(x), because of convexity of f,but can not prove that F(x) is convex.