  We show that, if a building is endowed with its complete system of
apartments, and if each panel is contained in at least four chambers, then the
intersection of two apartments can be any convex subcomplex contained in an
apartment. This combinatorial result is particularly interesting for lower
dimensional convex subcomplexes of apartments, where we definitely need the
assumption on the four chambers per panel in the building. The corresponding
statement is not true anymore for arbitrary systems of apartments, and
counter-examples for infinite convex subcomplexes exist for any type of
buildings. However, when we restrict to finite convex subcomplexes, the above
remains true for arbitrary systems of apartments if and only if every finite
subset of chambers of the standard Coxeter complex is contained in the convex
hull of two chambers.
