Geodesic
On tilings related to discrete reflection groups
Izvestiya. Mathematics , Tome 73 (2009) no. 6, pp. 1101-1109

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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.
Keywords: discrete reflection group, fundamental chamber, dual cone, Brouwer fixed-point theorem. 
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     author = {P. V. Bibikov and V. S. Zhgoon},
     title = {On tilings related to discrete reflection groups},
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P. V. Bibikov; V. S. Zhgoon. On tilings related to discrete reflection groups. Izvestiya. Mathematics , Tome 73 (2009) no. 6, pp. 1101-1109. http://geodesic.mathdoc.fr/item/IM2_2009_73_6_a1/