Determination of Boolean models by mean values of mixed volumes

TL;DR

This paper proves that mean values of mixed volumes in stationary Boolean models uniquely determine the underlying Poisson process in all dimensions, correcting previous incomplete proofs and extending results to four dimensions.

Contribution

It provides a complete proof for the 4-dimensional case and establishes a general uniqueness theorem using flag measures and valuation theory.

Findings

Corrected and completed the 4D mean volume determination

Established a general uniqueness theorem in all dimensions

Linked mixed volume formulas with valuation theory

Abstract

In Weil (2001) formulas were proved for stationary Boolean models $Z$ in $\mathbb{R}^d$ with convex or polyconvex grains, which express the densities of mixed volumes of $Z$ in terms of related mean values of the underlying Poisson particle process $X$. These formulas were then used to show that in dimensions 2 and 3 the mean values of $Z$ determine the intensity $\gamma$ of $X$. For $d=4$ a corresponding result was also stated, but the proof given was incomplete, since in the formula for the mean Euler characteristic $\overline V_0 (Z)$ a term $\overline V^{(0)}_{2,2}(X,X)$ was missing. This was pointed out in Goodey and Weil (2002), where it was also explained that a new decomposition result for mixed volumes and mixed translative functionals would be needed to complete the proof. Such a general decomposition result is now available based on flag measures of the convex bodies involved (Hug, Rataj and Weil (2013, 2017)). Here, we show that such flag representations not only lead to a correct derivation of the 4-dimensional result but even yield a corresponding uniqueness theorem in all dimensions. In the proof of the latter, we make use of Alesker's representation theorem for translation invariant valuations. We also discuss which shape information can be obtained in this way and comment on the situation in the non-stationary case.