On tilings related to discrete reflection groups

We get simpler proofs of theorems of Waldspurger and Meinrenken on tilings formed by sets of the form $(1-w)C^\circ$, $w\in W$, where $W$ is a linear or affine Weyl group and $C^\circ$ is the open kernel of a fundamental chamber $C$ of $W$. We also generalize these results to cocompact hyperbolic reflection groups.