Exercise 20, § 1

Let A be a building and S a set. Then A is said to be numbered by S if one is given a map $f$ from A to S such that, for any chamber C of A, the restriction of $f$ to C is a bijection from C to S. If F is a facet of A, $f(F)$ is called the type of F. Let A be a numbered building. An endomorphism $\varphi$ of A is called allowed if $a$ and $\varphi(a)$ are of the same type for all $a \in A$.

a) Let $\varphi$ be an endomorphism of A. Show that, if there exists a chamber C of A such that $a$ and $\varphi(a)$ are of the same type for all $a \in C$, then $\varphi$ is allowed. Show that if A is an apartment and $\pi$ is a folding of A (Exerc. 18), then $\pi$ is an allowed endomorphism.

b) A subset D of A is called convex if, for all $a \in A - D$, there exists an allowed endomorphism $\varphi$ of A such that $\varphi(x) = x$ for all $x \in D$ and $\varphi(a) \neq a$. Show that any intersection of convex sets is convex and that, for any subset D of A, there exists a smallest convex subset containing D: this is called the convex hull of D and denoted by $\Gamma(D)$.