Exercise 21, § 5

a) In $\mathbf{R}^n$ let $a_i$ ($1 \leq i \leq n + 1$) be $n + 1$ points that are affinely independent. Denote the convex envelope of the $a_i$ by S and the convex envelope of the $a_i$ with $i \neq k$ by $F_k$ for $1 \leq k \leq n + 1$. For each k let $C_k$ be a compact convex set containing $F_k$, and suppose that S is contained in the union of the $C_k$; show then that $\bigcap_{k=1}^{n+1} C_k \neq \varnothing$. (Argue by reductio ad absurdum and induction on $n$, considering the intersection $C'_{n+1}$ of the $C_i$ with indices $i \leq n$ and supposing that $C_{n+1} \cap C'_{n+1} = \varnothing$, which would allow the strict separation of the two convex sets by a hyperplane.) b) Let X be a compact convex set in a Hausdorff topological vector space E, and $(C_\lambda)_{\lambda \in L}$ a family of compact convex sets contained in X, such that for every set $H \subset L$ having $n$ (resp. m) elements, the intersection (resp. the union) of the $C_\lambda$ with indices $\lambda \in H$ is not empty (resp. is equal to X). Show that if $m \leq n + 1$ the intersection $\bigcap_{\lambda \in L} C_\lambda$ is not empty. (This is effectively proving that for any finite set H of $p \geq m$ indices of L, we have $\bigcap_{\lambda \in H} C_\lambda \neq \varnothing$. Argue by induction on $p$ assuming that the result has been proved for $p - 1$ indices. Argue then by reductio ad absurdum, considering for each index $i \in H$ a point $a_i \in \bigcap_{\lambda \in H - \{i\}} C_\lambda$, and showing by the aid of Helly’s th. (II, p. 68, exerc. 21) that the $a_i$ generate a linear variety of dimension $p - 1$, then finally apply a).)