Exercise 19, § 7

In the space $\mathbf{R}^n$, let $(K_\alpha)$ be a family of compact sets, in number $\geq n + 1$, and such that none of them is contained in an affine hyperplane. Suppose that for every family $(u_\alpha)$ of affine automorphisms of $\mathbf{R}^n$, if any $n + 1$ of the sets $u_\alpha(K_\alpha)$ have a common point, then all the $u_\alpha(K_\alpha)$ have a common point. Show that under these conditions the $K_\alpha$ are convex. (Suppose on the contrary that there exist $n + 1$ points $x_1, ..., x_{n+1}$ in the same set $K_\alpha$ and a point $x_0$ which belongs to the convex envelope of the set of the $x_i$ ($i \geq 1$) but does not belong to $K_\alpha$. Note that for every index $i \geq 1$, there is an affine automorphism $u_i$ of $\mathbf{R}^n$ and an index $\alpha_i$ such that $x_0$ and the $x_j$ of index $j \neq i$ are extremal points of $u_i(K_{\alpha_i})$, and show that the $n + 2$ sets $K_\alpha$ and $u_i(K_{\alpha_i})$ have no points in common.)