Integral Geometry of pairs of planes

TL;DR

This paper explores integral geometry related to pairs of planes in three-dimensional space, expressing certain integrals via visual angles of convex sets and evaluating inequalities in Crofton-type formulas.

Contribution

It introduces new expressions for integrals of invariant measures of plane pairs using visual angles and assesses the Crofton inequality deficit.

Findings

Derived formulas linking plane pair integrals to convex set visual angles

Evaluated the Crofton-type inequality deficit in specific geometric contexts

Extended integral geometric methods to new classes of convex sets

Abstract

We deal with integrals of invariant measures of pairs of planes in euclidean space $\mathbb{E}^3$ as considered by Hug and Schneider. In this paper we express some of these integrals in terms of functions of the visual angle of a convex set. As a consequence of our results we evaluate the deficit in a Crofton-type inequality due to Blashcke.