Ryll-Nardzewski fixed-point theorem

Remarks

The isometry condition can be weakened to the condition that $S$ is noncontracting: for $x \ne y$ in $K$, the vector $0$ is not in the norm closure of $\set{s(x) - s(y) : s \in S}$. In this form the theorem holds for locally convex spaces and weakly continuous affine maps. By weak compactness of $K$, it suffices to treat finitely generated semigroups. For these, one takes a fixed point of a suitable average of the generators. The noncontracting condition and a geometric lemma on weakly compact convex sets show that it is fixed by each generator.

The isometry hypothesis cannot be dropped. On $K = [0, 1]$, the constant maps $s_0(x) = 0$ and $s_1(x) = 1$ are affine and generate the semigroup $\set{s_0, s_1}$, but they have no common fixed point.

Example: let $G$ be the symmetric group on $\set{1, \ldots, n}$, acting on the simplex $K = \set{x \in \R^n : x_i \ge 0, \sum x_i = 1}$ by permuting coordinates. These maps are affine isometries for the Euclidean norm, and the common fixed point is the barycentre $(1/n, \ldots, 1/n)$.

Going deeper

For a student

A classical application, in the locally convex form, is the existence of Haar measure on a compact group. The group acts by affine maps on a compact convex set of means on the continuous functions. A common fixed point is an invariant mean, which gives the Haar integral. The theorem should be compared with the Markov–Kakutani theorem, which asks for commuting continuous affine maps instead of isometries.