Fan Valuations and Spherical Intrinsic Volumes

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 sphe...

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Bibliographic Details
Published in:Annals of combinatorics 2024-12, Vol.28 (4), p.1285-1302
Main Authors: Backman, Spencer, Manecke, Sebastian, Sanyal, Raman
Format: Article
Language:English
Subjects:
Combinatorics
Cones
Deletion
Hyperplanes
Invariants
Mathematics
Mathematics and Statistics
Original Paper
Polynomials
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Summary: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.
ISSN:0218-0006
0219-3094
DOI:10.1007/s00026-024-00699-x