Proposition 2

The set of facets is locally finite.

This is clear since $\mathscr{H}$ is locally finite.

Let F be a facet and a a point of F. A hyperplane $H \in \mathscr{H}$ contains F if and only if $a \in H$; the set $\mathscr{F}$ of these hyperplanes is thus finite; their intersection is an affine subspace L of E, which we shall call the affine support of F; the dimension of L will be called the dimension of F.

If $\mathscr{N}$ is the set of hyperplanes $H \in \mathscr{H}$ not containing F, then

We shall prove that the closure of F is given by

It is clear that the right-hand side contains the left-hand side. Conversely, let $x \in L \cap \bigcap_{H \in \mathscr{N}} \overline{D_H(a)}$. The open segment with extremities a and x is contained in L and in each of the $D_H(a)$ for $H \in \mathscr{N}$, and hence in F. It follows that x is in the closure of F, hence the formula.