Abstract
We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones are given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans that this held for all finite real reflection arrangements. The methods used are geometric and combinatorial. As a consequence, we determine that the angle sums of a zonotope are given by the characteristic polynomial of the order dual of the intersection lattice of the arrangement.
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The work here was done while the first author was a visiting scholar in the mathematics department at Cornell University. The second author is partially supported by NSF grant DMS-0900912.
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Klivans, C.J., Swartz, E. Projection Volumes of Hyperplane Arrangements. Discrete Comput Geom 46, 417–426 (2011). https://doi.org/10.1007/s00454-011-9363-7
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DOI: https://doi.org/10.1007/s00454-011-9363-7