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arXiv · 2106.06407

Fan Valuations and spherical intrinsic volumes

Spencer Backman·Sebastian Manecke·Raman Sanyal
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Abstract

We generalize valuations on polyhedral cones to valuations on 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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Spencer Backman, Sebastian Manecke, Raman Sanyal. 2021-06-11. Fan Valuations and spherical intrinsic volumes. https://arxiv.org/abs/2106.06407

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