Exercise 34, § 2

Let U, V be two open convex sets in a Hausdorff topological vector space E such that $\overline{V} \subset U$ and that U does not contain any half line. Let $\mathscr{F}$ be a set of convex functions defined in $\overline{U}$, uniformly bounded above on the frontier of U and uniformly bounded below on the frontier of V. Show that $\mathscr{F}$ is equicontinuous.