The Klain approach to zonal valuations

TL;DR

This paper extends the Klain-Schneider theorem to zonal valuations invariant under rotations around a fixed axis, providing new integral representations and geometric formulas, simplifying existing proofs and broadening applications.

Contribution

It introduces a new integral representation for zonal valuations, simplifies proofs of existing theorems, and extends integral geometric formulas for zonal cases.

Findings

New integral representation of zonal valuations involving mixed area measures

Simplified proof of the characterization of zonal valuations by Knoerr

Extended zonal integral geometric formulas

Abstract

We show an analogue of the Klain-Schneider theorem for valuations that are invariant under rotations around a fixed axis, called zonal. Using this, we establish a new integral representation of zonal valuations involving mixed area measures with a disk. In our argument, we introduce an easy way to translate between this representation and the one involving area measures, yielding a shorter proof of a recent characterization by Knoerr. As applications, we obtain various zonal integral geometric formulas, extending results by Hug, Mussnig, and Ulivelli. Finally, we provide a simpler proof of the integral representation of the mean section operators by Goodey and Weil.