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Fan Valuations and Spherical Intrinsic Volumes

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Abstract

We generalize valuations on polyhedral cones to valuations on (plane) fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion– restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion–restriction proof of a result of Klivans–Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these intrinsic indicators yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.

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Acknowledgements

Work on this project started at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester on Geometric and Topological Combinatorics. The authors acknowledge support by National Science Foundation under Grant No. DMS-1440140 during our stay at the MSRI as well as from the DFG-Collaborative Research Center, TRR 109 “Discretization in Geometry and Dynamics”. The first author was supported by a Simons Collaboration Gift No. 854037 and NSF Grant (DMS-2246967). The authors thank the referee for suggestions that helped improve the exposition.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Vermont, Burlington, VT, USA

    Spencer Backman

  2. Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt, Germany

    Sebastian Manecke & Raman Sanyal

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  1. Spencer Backman
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  2. Sebastian Manecke
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Correspondence to Spencer Backman.

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Backman, S., Manecke, S. & Sanyal, R. Fan Valuations and Spherical Intrinsic Volumes. Ann. Comb. (2024). https://doi.org/10.1007/s00026-024-00699-x

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