Units, polyhedra, and a conjecture of Satake

Let $F/\QQ $ be a totally real number field of degree $n$. We explicitly evaluate a certain sum of rational functions over a infinite fan of $F$-rational polyhedral cones in terms of the norm map $\Norm \colon F\to \QQ $. This completes Sczech's combinatorial proof of Satake's conjecture connecting the special values of $L$-series associated to cusp singularities with intersection numbers of divisors in their toroidal resolutions.