Proposition 10

*Let C be a chamber, let H and H' be two walls of C, and let L be a hyperplane meeting $D_H(C) \cap D_{H'}(C)$. Assume that H is distinct from H' and that one of the following conditions is satisfied:

a) The hyperplanes H, H' and L are parallel.

b) The hyperplanes H and H' are not parallel, and $\overline{H \cap H'} \subset L$.

Then, L meets C.

Let $b$ (resp. $b'$) be a point of the face of C with support H (resp. $H'$); it is immediate that every point of the segment $[bb']$ distinct from $b$ and $b'$ belongs to C.

Introduce an affine function $f$ that vanishes at every point of H and is such that $f(x) > 0$ for $x$ in $D_H(C)$; similarly, introduce an affine function $f'$ having an analogous property with respect to $H'$. By applying Lemma 2, we can find numbers $\lambda$ and $\lambda'$ and an affine function $g$ having the properties stated in the lemma. We have $(\lambda, \lambda') \neq (0, 0)$, and for every point $x$ of $L \cap D_H(C) \cap D_{H'}(C)$, we have $f(x) > 0,\ f'(x) > 0$ and $\lambda.f(x) + \lambda'.f'(x) = 0$, so $\lambda \lambda' < 0$. On the other hand, we have $g(b) = \lambda'.f'(b)$ and $g(b') = \lambda.f(b')$, and since $f(b') > 0,\ f'(b) > 0$, we have $g(b)g(b') < 0$. The points $b$ and $b'$ are thus strictly on opposite sides of the hyperplane L, hence there exists a point c of L which belongs to $[bb']$, and is distinct from $b$ and $b'$, hence that belongs to C.