A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits 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 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 […] 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?

A crucial aspect of the Bruhat–Tits theory of affine buildings is the Bruhat–Tits 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?