  In this paper we study convex subcomplexes of spherical buildings. We pay
special attention to fixed point sets of type-preserving isometries of
spherical buildings. This sets are also convex subcomplexes of the natural
polyhedral structure of the building. We show, among other things, that if the
fixed point set is top-dimensional then it is either a subbuilding or it has
circumradius $\leq \frac{\pi}{2}$. If the building is of type $A_n$ or $D_n$,
we also show that the same conclusion holds for an arbitrary (top-dimensional
in the $D_n$-case) convex subcomplex. This proves a conjecture of Kleiner-Leeb
in these cases.
