Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 2 / c / Solution

By part (b), the hypothesis says that every wall separates $e$ from $w_0$. Let $u\in W$. For each wall, $e$ and $w_0$ lie in opposite half-spaces, so $u$ lies on exactly one of their two sides. The wall therefore separates exactly one of the pairs $(e,u)$ and $(u,w_0)$. Since Coxeter length equals the number of separating walls,