What is Ryll-Nardzewski fixed-point theorem?

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if E {\displaystyle E} is a normed vector space and K {\displaystyle K} is a nonempty convex subset of E {\displaystyle E} that is compact under the weak topology, then every group of affine isometries of K {\displaystyle K} has at least one fixed point.

This theorem was announced by Czesław Ryll-Nardzewski. Later Namioka and Asplund gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit.