Rotational Crofton formulae for Minkowski tensors and some affine counterparts

TL;DR

This paper develops new rotational Crofton formulas for Minkowski tensors, linking their averages over sections to the geometry of the set, with explicit formulas involving hypergeometric functions for various subspace dimensions.

Contribution

It introduces novel rotational Crofton formulas for Minkowski tensors, including explicit hypergeometric function expressions, extending the geometric analysis of sets of positive reach.

Findings

Derived explicit formulas for rotational averages involving hypergeometric functions.

Extended Crofton formulas to affine subspaces not passing through the origin.

Provided detailed analysis for sections with lines and hyperplanes.

Abstract

Motivated by applications in local stereology, a new rotational Crofton formula is derived for Minkowski tensors. For sets of positive reach, the formula shows how rotational averages of intrinsically defined Minkowski tensors on sections passing through the origin are related to the geometry of the sectioned set. In particular, for Minkowski tensors of order j-1 on j-dimensional linear subspaces, we derive an explicit formula for the rotational average involving hypergeometric functions. Sectioning with lines and hyperplanes through the origin is considered in detail. We also study the case where the sections are not restricted to pass through the origin. For sets of positive reach, we here obtain a Crofton formula for the integral mean of intrinsically defined Minkowski tensors on j-dimensional affine subspaces.