Fan Valuations and spherical intrinsic volumes

TL;DR

This paper extends valuations from polyhedral cones to fans, introduces a characteristic polynomial based on spherical intrinsic volumes, and connects these concepts to hyperplane arrangements and existing invariants.

Contribution

It generalizes valuations to fans, defines a new characteristic polynomial, and links spherical intrinsic volumes with deletion-restriction invariants in hyperplane arrangements.

Findings

Defined a characteristic polynomial for fans matching hyperplane arrangements.

Established a correspondence between valuations and deletion-restriction invariants.

Generalized projection volume results to hyperplane arrangements.

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.