Exercise 29, § 5

Let $X$ be a compact convex set in a Hausdorff locally convex space $E$. Let $u$ be a lower semi-continuous convex function in $X$ and $v$ an upper semi-continuous concave function in $X$ such that $u(x) > v(x)$ for all $x \in X$. Then there exists an affine linear function $f$ that is continuous in $E$ and such that $v(x) < f(x) < u(x)$ for all $x \in X$.