Exercise 1, Appendix 0

Let E be a Hausdorff locally convex space, K a compact convex subset of E, and S a set of continuous affine linear transformations from K into itself, which is stable under composition. The set S is said to be distal if, for every pair of distinct points a, b in K, the closure of the set of pairs (s.a, s.b), where s ranges over S, does not contain any point of the diagonal of K × K. a) Show that an equicontinuous group of affine transformations of K is distal. b) Show that if K is non-empty and S is distal, then there exists at least one point of K which is invariant under every transformation of S. (If M is a non-empty compact convex subset of K which is stable under S, show that if M contains two distinct points x_1, x_2, and if A is the closure of the orbit of $x = \frac{x_1 + x_2}{2}$, then A cannot contain any extremal point of M. Deduce that if L is a minimal element of the family of non-empty compact convex subsets of K, which are stable under S, then L reduces to a point; argue by reductio ad absurdum: with the same notations, the closed convex envelope of A would be equal to L, which would contradict the Krein-Milman theorem.)