Exercise 28, § 5

Let $X$ be a compact convex subset of $E$ a Hausdorff locally convex space, and let $f$ be an upper semi-continuous convex function in $X$. Show that the set $L$ of continuous convex functions $g$ in $X$ such that $g(x) > f(x)$ for all $x \in X$ is decreasing directed and that its lower envelope is equal to $f$. (Let $u, v$ be elements of $L$. To construct an element of $L$ which is less than $u$ and $v$, use reasoning analogous to that of prop. 6, II, p. 40. Interpret the set $K_1$ analogous to the set $K$ in this argument as the set of points situated above the graph of a lower semi continuous function that is less than $u$ and $v$ and strictly larger than $f$ at every point; apply prop. 5 of II, p. 39 and Dini’s th. to this function. To show that the lower envelope of $L$ is $f$, note that $f$ is bounded above by a constant $b$; $(x, t)$ being a point of $E \times \mathbf{R}$ situated above the graph of $f$, let $K'$ be the convex envelope of $\{ (x, t) \} \cup (X' \times \{ b \})$, where $X'$ is a convenient compact neighbourhood of $x$ in $X$; argue with $K'$ as above for $K_1$.)