Exercise 39, § 7

Let $H$ be a closed hyperplane in a Hausdorff locally convex space $E$, that does not contain 0 and let $S$ be a compact simplex contained in $H$ (II, p. 71, exerc. 41). a) Let $C$ be the cone with vertex 0 generated by $S$. Show that if $(x_i)_{i \in I}$ and $(y_j)_{j \in J}$ are two finite families of points of $C$ such that $\sum x_i = \sum y_j$, then there exists a finite family $(z_{ij})_{(i,j) \in I \times J}$ of points of $C$ such that $x_i = \sum_{j \in J} z_{ij}$ for all $i \in I$ and $y_j = \sum_{i \in I} z_{ij}$ for all $j \in J$ (argue by induction to reduce to the case $I = J = \{1, 2\}$). b) Let $f$ be a convex, upper semi-continuous function in $S$. Show that the function $\tilde{f}$ (defined in exerc. 38) is an affine function. (First reduce to the case where $f$ is continuous by using exerc. 38, b) and II, p. 81, exerc. 28. Next use the fact if $f$ is continuous, then $\tilde{f} = f'$ (exerc. 38, e)), and show that $f'$ is convex using a) to bound $f'(\alpha_1 x_1 + \alpha_2 x_2)$ above when $\alpha_1 \geq 0, \alpha_2 \geq 0, \alpha_1 + \alpha_2 = 1$.)