The Klain approach to zonal valuations

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 integral geometric formulas for SO(n − 1): an additive kinematic, a Kubota-, and a Crofton-type formula. This extends 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.