Crofton Formulas and Indefinite Signature

TL;DR

This paper establishes that all $O(p,q)$-invariant valuations can be represented by Crofton formulas, providing explicit formulas for certain signatures and degrees, and analyzing the associated invariant distributions and their properties.

Contribution

It proves that every $O(p,q)$-invariant valuation admits an $O(p,q)$-invariant Crofton formula, extending explicit formulas to general signatures and identifying invariant formulas for all $O(p,2)$-invariant valuations.

Findings

Every $O(p,q)$-invariant valuation is given by an invariant Crofton formula.

Explicit Crofton formulas are obtained for specific signatures and degrees, including $O(p,p)$ and $O(p,2)$.

The invariant Crofton distributions are explicitly characterized, and their properties are analyzed.

Abstract

We study the $O(p,q)$-invariant valuations classified by A. Bernig and the author. Our main result is that every such valuation is given by an $O(p,q)$-invariant Crofton formula. This is achieved by first obtaining a handful of explicit formulas for a few sufficiently general signatures and degrees of homogeneity, notably in the $(p-1)$ homogeneous case of $O(p,p)$, yielding a Crofton formula for the centro-affine surface area when $p\not\equiv 3\mod 4$. We then exploit the functorial properties of Crofton formulas to pass to the general case. We also identify the invariant formulas explicitly for all $O(p,2)$-invariant valuations. The proof relies on the exact computation of some integrals of independent interest. Those are related to Selberg's integral and to the Beta function of a matrix argument, except that the positive-definite matrices are replaced with matrices of all signatures. We also analyze the distinguished invariant Crofton distribution supported on the minimal orbit, and show that, somewhat surprisingly, it sometimes defines the trivial valuation, thus producing a distribution in the kernel of the cosine transform of particularly small support. In the heart of the paper lies the description by Muro of the $|\det X|^s$ family of distributions on the space of symmetric matrices, which we use to construct a family of $O(p,q)$-invariant Crofton distributions. We conjecture there are no others, which we then prove for $O(p,2)$ with $p$ even. The functorial properties of Crofton distributions, which serve an important tool in our investigation, are studied by T. Wannerer and the author in the Appendix.