Lefschetz operators on convex valuations
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929253000413184 |
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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 investigate the action of Alesker's Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain-Schneider functions by a Radon type transform, generalizing a result by Schuster and Wannerer. In the case of rotationally equivariant Minkowski valuations, the Lefschetz operators act on the generating function as a convolution transform. We show that the convolution kernel satisfies a Legendre type differential equation, and thus, is a strictly positive function that is smooth up to one point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14731 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lefschetz operators on convex valuations Brauner, Leo Hofstätter, Georg C. Ortega-Moreno, Oscar Metric Geometry 52B45, 52A39, 52A20, 44A12, 44A35 We investigate the action of Alesker's Lefschetz operators on translation invariant valuations on convex bodies. For scalar valued valuations, we describe this action on the level of Klain-Schneider functions by a Radon type transform, generalizing a result by Schuster and Wannerer. In the case of rotationally equivariant Minkowski valuations, the Lefschetz operators act on the generating function as a convolution transform. We show that the convolution kernel satisfies a Legendre type differential equation, and thus, is a strictly positive function that is smooth up to one point. |
| title | Lefschetz operators on convex valuations |
| topic | Metric Geometry 52B45, 52A39, 52A20, 44A12, 44A35 |
| url | https://arxiv.org/abs/2402.14731 |