Spherical subcomplexes of spherical buildings

Abstract

Let $\Delta$ be a thick, spherical building equipped with its natural CAT(1) metric and let $M$ be a proper, convex subset of $\Delta$. If $M$ is open or if $M$ is a closed ball of radius $\pi ∕2$, then $\Lambda$, the maximal subcomplex supported by $\Delta \setminus M$, is $dim\Lambda$–spherical and non-contractible.