Ryll-Nardzewski fixed-point theorem

In functional analysis, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if $E$ is a normed vector space and $K$ is a nonempty convex subset of $E$ that is compact under the weak topology, then every group (or equivalently: every semigroup) of affine isometries of $K$ has at least one fixed point. (Here, a fixed point of a set of maps is a point that is fixed by each map in the set.)

This theorem was announced by Czesław Ryll-Nardzewski.{{cite journal|first=C.|last=Ryll-Nardzewski|title=Generalized random ergodic theorems and weakly almost periodic functions|journal=Bull. Acad. Polon. Sci. Sér. Sci. Math. Astron. Phys.|volume=10|year=1962|pages=271–275}} Later Namioka and Asplund {{cite journal|doi=10.1090/S0002-9904-1967-11779-8|first=I.|last=Namioka|author1-link= Isaac Namioka |author2=Asplund, E. |title=A geometric proof of Ryll-Nardzewski's fixed point theorem|journal=Bull. Amer. Math. Soc.|volume=73|issue=3|year=1967|pages=443–445|doi-access=free}} gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit.{{cite journal|first=C.|last=Ryll-Nardzewski|title=On fixed points of semi-groups of endomorphisms of linear spaces|journal=Proc. 5th Berkeley Symp. Probab. Math. Stat|volume=2: 1|publisher=Univ. California Press|year=1967|pages=55–61}}