Brauner, L., Hofstätter, G. C., & Ortega Moreno, O. A. (2024). The Klain approach to zonal valuations. arXiv. https://doi.org/10.48550/arXiv.2410.18651
valuations; Klain-Schneider theorem; convex geometry; mixed area measures; integral geometry
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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.
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Project title:
Bewertungen auf konvexen Funktionen: P 34446-N (FWF - Österr. Wissenschaftsfonds) Fixpunkt Probleme und isoperimetrische Ungleichungen: ESP 236-N (FWF - Österr. Wissenschaftsfonds)
