Integral geometry of translation invariant functionals, II: The case of general convex bodies

TL;DR

This paper extends translative integral formulas for translation invariant functionals on convex bodies, with applications to stochastic geometry models like Poisson processes and Boolean models, including new results for flag measures.

Contribution

It generalizes translative formulas to all continuous translation invariant valuations on convex bodies and explores their applications in stochastic geometry.

Findings

Translative formulas extend to all continuous translation invariant valuations.

New results obtained for flag measures in Poisson and Boolean models.

Applications demonstrate the utility of these formulas in stochastic geometry.

Abstract

In continuation of Part I, we study translative integral formulas for certain translation invariant functionals, which are defined on general convex bodies. Again, we consider local extensions and use these to show that the translative formulas extend to arbitrary continuous and translation invariant valuations. Then, we discuss applications to Poisson particle processes and Boolean models which contain, as a special case, some new results for flag measures.