Rotational Crofton Formulae with a Fixed Subspace

TL;DR

This paper generalizes Crofton formulae to include sections containing a fixed subspace, providing new integral relations for intrinsic volumes with potential stereological applications.

Contribution

It introduces a new class of Crofton formulae constrained to fixed subspaces, extending classical rotational formulas and proving their optimality.

Findings

Generalized Crofton formula for fixed subspace sections

Proof combining Blaschke--Petkantschin and classical Crofton formulas

Established integral relation for vertical sections

Abstract

The classical Crofton formula explains how intrinsic volumes of a convex body $K$ in $n$-dimensional Euclidean space can be obtained from integrating a measurement function at sections of $K$ with invariantly moved affine flats. Motivated by stereological applications, we present variants of Crofton's formula, where the flats are constrained to contain a fixed linear subspace $L_0$, but are otherwise invariantly rotated. This main result generalizes a known rotational Crofton formula, which only covers the case $\dim L_0=0$. The proof combines a suitable Blaschke--Petkantschin formula with the classical Crofton formula. We also argue that our main result is best possible, in the sense that one cannot estimate intrinsic volumes of a set, based on lower-dimensional sections, other than those given by our result. Finally, we provide a proof for a well-established variant: an integral relation for vertical sections. Our formula is stated for intrinsic volumes of a given set, complementing the classical approach for Hausdorff measures.