Exercise 21, § 2

a) Let $A_i$ ($1 \leq i \leq r$) $r > n + 1$, be a family of convex subsets of $\mathbf{R}^n$ such that any $r - 1$ of the $A_i$ have a non-empty intersection; show that the r sets $A_i$ have a non-empty intersection (Helly's theorem). (Let $x_i$ be a point of the intersection of the $A_j$ with indexes $j \neq i$; there exist r numbers $\lambda_i$ which are not all zero and are such that $\sum_{i=1}^r \lambda_i = 0$ and $\sum_{i=1}^r \lambda_i x_i = 0$; in this last equation take to one side those terms with $\lambda_i \geq 0$ and to the other those with $\lambda_i < 0$.) b) Given a family of compact convex sets in $\mathbf{R}^n$, show that the intersection of all the sets of the family is non-empty if the intersection of any selection of $n + 1$ sets of the family is non-empty. c) In $\mathbf{R}^n$, let K be a convex set and $(A_i)_{1 \leq i \leq r}$ be a family of $r > n + 1$ convex sets. Suppose that for every selection of $n + 1$ indices $(i_k)$ each less than or equal to r there exists $a \in \mathbf{R}^n$ such that $a + K$ contains each of the $A_{i_k}$. Show that then there exists $b \in \mathbf{R}^n$ such that $b + K$ contains all the $A_i$. Show that similar results hold if « contains » is replaced by « is contained in » or by « meets in a non-empty set ». (For each index i, consider the set $C_i$ of the $x \in \mathbf{R}^n$ for which $x + K \supset A_i$ (or $x + K \subset A_i$, or $(x + K) \cap A_i \neq \varnothing$)). Generalize to any family of compact convex sets of $\mathbf{R}^n$.

EXERCISES