The Klain approach to zonal valuations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912089282445312 |
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| author | Brauner, Leo Hofstätter, Georg C. Ortega-Moreno, Oscar |
| author_facet | Brauner, Leo Hofstätter, Georg C. Ortega-Moreno, Oscar |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_18651 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Klain approach to zonal valuations Brauner, Leo Hofstätter, Georg C. Ortega-Moreno, Oscar Metric Geometry Functional Analysis 52B45, 52A39, 52A20 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. |
| title | The Klain approach to zonal valuations |
| topic | Metric Geometry Functional Analysis 52B45, 52A39, 52A20 |
| url | https://arxiv.org/abs/2410.18651 |