If $L$ is a closed convex subset of compact convex set $K$, then the restriction map $A(K)\to A(L)$ is surjective

Question

For a convex set $X$ denote by $A(X)$ the set of continuous affine functions on $X$.

Let $L$ be a closed convex subset of compact Hausdorff convex set $K$, then I want to show that the restriction map $A(K)\to A(L)$ is surjective.

If we just considered continuous functions, then the restriction map $C(K)\to C(L)$ is really a surjective homomorphism (one can use for example Tietze extension Theorem).

Thanks for any help