A crucial aspect of the Bruhat–Tits theory of affine buildings is the [Bruhat–Tits fixed-point theorem](https://en.wikipedia.org/wiki/Bruhat%E2%80%93Tits_fixed_point_theorem), which, in one of many formulations, states that, if $\Gamma$ is a group of isometries of an affine building and $S$ is a closed, bounded, convex, $\Gamma$-stable subset of the affine building, then $\Gamma$ admits a fixed point on $S$.


Is there any similar result for spherical buildings (specifically of spherical buildings attached to semisimple groups, in case there are more results for them than for general spherical buildings)? I am particularly interested in results of the form: if $\Gamma$ is a group of isometries of a spherical building and $S$ is a […] $\Gamma$-stable subset of the spherical building, then $\Gamma$ admits a fixed point on $S$. For example, does this hold if we require $S$ to be closed and convex? (Probably not.) Does it hold if we require $S$ also to be contractible, or perhaps just never to contain two opposite simplices?



Fixed points on spherical buildings
Since spherical buildings are CAT(1), we get a fixed point if $$\mathop{\rm rad}S<\tfrac \pi 2.$$


