Crofton formulae for products

TL;DR

This paper develops new integral-geometric Crofton formulae generalizing classical results by incorporating mixed Riemannian volumes, based on advanced calculations in the ring of normal densities.

Contribution

It introduces generalized Crofton formulae using mixed Riemannian volumes, expanding the scope of integral geometry techniques.

Findings

Derived new Crofton formulae involving mixed Riemannian volumes

Extended classical Crofton formulae to more general geometric contexts

Utilized the ring of normal densities for advanced calculations

Abstract

It is shown how new integral-geometric formulae can be obtained from the existing formulae of Crofton type. In particular, for classical Crofton formulae in which the answer depends on the Riemannian volume, we obtain generalizations in terms of the mixed Riemannian volume defined in the paper. The method is based on the calculations in the ring of normal densities constructed in the previous work of the authors.