Exercise 38, § 7

Let E be a Hausdorff locally convex space and X a compact convex set in E. Denote by $\mathscr{A}(X)$ the set of continuous affine functions in X (not necessarily restrictions to X of continuous affine functions in E, cf. II, p. 78, exerc. 11). For every real valued function f that is bounded above in X, put $\tilde{f}(x) = \inf_h(h(x))$ for all x ∈ X where h varies in the set of function of $\mathscr{A}(X)$ such that $h \geq f$. a) Show that $\tilde{f}$ is an upper semi-continuous concave function. If $f$ itself is upper semicontinuous and concave, then $\tilde{f} = f$ (cf. II, p. 39, prop. 5).

b) Suppose that $f$ is upper semi-continuous. Show that $\tilde{f}$ is the lower envelope of the function $\tilde{g}$, when $g$ varies in a set of functions that are continuous in $X$ of which $f$ is the lower envelope. c) For all $x \in X$, write $\mathscr{M}_x$ for the set of finite families $\mu = ((\mu_j, x_j))$ where the $x_j$ are points of $X$, the $\mu_j$ are numbers $\geqslant 0$ satisfying $\sum \mu_j = 1$, such that $x = \sum \mu_j x_j$. For each real valued function $f$ bounded above in $X$, put $f'(x) = \sup_{\mu \in \mathscr{M}_x} \sum \mu_j f(x_j)$ for all $x \in X$. Show that $f'$ is a concave function in $X$ and that $f' \leqslant \tilde{f}$. d) Suppose that $f$ is continuous in $X$. Given $\varepsilon > 0$, let $(U_k)_{1 \leq k \leq N}$ be a covering of $X$ by convex open sets such that the relations $x \in U_k, y \in U_k$ imply the inequality $|f(x) - f(y)| \leq \varepsilon$. Put $A_1 = U_1$ and, for $k > 1$, $A_k = U_k \cap (\cup_{i=1}^{k-1} U_i)$. Show that, for all $x \in X$, there exists a family $\mu = ((\mu_k, x_k))$ of N terms belonging to $\mathscr{M}_x$ with $x_k \in U_k$ for $1 \leq k \leq N$ such that $\sum_k \mu_k f(x_k) \geq f'(x) - 2\varepsilon$ (if we have the inequality $\sum_j \lambda_j f'(y_j) \geq f'(x) - \varepsilon$, group the $y_j$ belonging to the same $A_k$ together). e) Deduce from d) that when $f$ is continuous then $f'$ is upper semi-continuous and $f' = \tilde{f}$. (If $\mathscr{U}$ is an ultrafilter on $X$ finer than the filter of the neighbourhoods of a point $x \in X$ and if $f'(y) \geq r$ for all the points $y$ of a set belonging to $\mathscr{U}$, show that $f'(x) \geq r - 2\varepsilon$, by making a family $\mu_y \in \mathscr{M}_y$ correspond to each $y$, which satisfies condition d) and proceeding to the limit through $\mathscr{U}$.)