On Continuity of Buffon Functionals in the Space of Planes in $\boldsymbol{\mathbb{R}}^{\mathbf{3}}$

Abstract

The paper considers measures in the space $\mathbb{E}$ of planes in $\mathbb{R}^{3}$, and combinatorial decompositions for their values on ‘‘Buffon sets’’ in $\mathbb{E}$. These decompositions, written in terms of a ‘‘wedge function’’ depending on the measure, have been known since long in Combinatorial Integral Geometry, yet their explicit expressions have been well established only for ‘‘non-degenerate’’ Buffon sets. Theorem 1 removes this gap and presents a decomposition algorithm valid with no similar restriction. Theorem 2 presents a result in a direction converse to Theorem 1. Starting from the decomposition algorithm, a combinatorial valuation $\Psi_{F}$ is defined that depends on ‘‘general’’ continuous additive wedge function $F(W)$. The question is: when $\Psi_{F}$ becomes a measure in the space $\mathbb{E}$? Theorem 2 points at special ‘‘tetrahedral inequalities’’, the analogues of triangular inequalities of the planar theory. If $\Psi_{F}$ satisfies these ‘‘tetrahedral inequalities’’, then $\Psi_{F}$ becomes a measure and the corresponding $F(W)$ is called a ‘‘wedge metric’’ (to stress the connection of the paper’s topic with Hilbert’s Fourth Problem).