Ryll-Nardzewski fixed-point theorem

In <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a branch of mathematics, the Ryll-Nardzewski fixed-point theorem states that if 
 
 
 
 E
 
 
 {\displaystyle E}
 
 is a <a href="/facts/Normed_vector_space/rmDljfyE">normed vector space</a> and 
 
 
 
 K
 
 
 {\displaystyle K}
 
 is a nonempty <a href="/facts/Convex_set/vdAuJRJl">convex</a> subset of 
 
 
 
 E
 
 
 {\displaystyle E}
 
 that is <a href="/facts/Compact_space/d0cgXJH7">compact</a> under the <a href="/facts/Weak_topology/XJASE9Up">weak topology</a>, then every <a href="/facts/Group_(mathematics)/IQTYqINt">group</a> (or equivalently: every <a href="/facts/Semigroup/rgSdk2kt">semigroup</a>) of <a href="/facts/Affine_map/BD7yDr10">affine</a> <a href="/facts/Isometry/K5wX8ah6">isometries</a> of 
 
 
 
 K
 
 
 {\displaystyle K}
 
 has at least one fixed point. (Here, a fixed point of a set of maps is a point that is <a href="/facts/Fixed_point_(mathematics)/vVVddTKc">fixed</a> by each map in the set.)
This theorem was announced by <a href="/facts/Czes%C5%82aw_Ryll-Nardzewski/Eny49wxk">Czesław Ryll-Nardzewski</a>. Later Namioka and Asplund gave a proof based on a different approach. Ryll-Nardzewski himself gave a complete proof in the original spirit.

Ryll-Nardzewski fixed-point theorem open-in-new

Ryll-Nardzewski fixed-point theorem