Steiner symmetrization on the sphere
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866910793539256320 |
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| author | Basit, Bushra Hoehner, Steven Lángi, Zsolt Ledford, Jeff |
| author_facet | Basit, Bushra Hoehner, Steven Lángi, Zsolt Ledford, Jeff |
| contents | The aim of this paper is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbolic space introduced by J. Schneider (Manuscripta Math. 60: 437-461, 1988). We show that this symmetrization preserves volume in every dimension, and convexity in the spherical plane, but not in dimensions $n > 2$. In addition, we investigate the monotonicity properties of the perimeter and diameter of a set under this process, and find conditions under which the image of a spherically convex disk under a suitable sequence of Steiner symmetrizations converges to a spherical cap. We apply our results to prove a spherical analogue of a theorem of Sas, and to confirm a conjecture of Besau and Werner (Adv. Math. 301: 867-901, 2016) for centrally symmetric spherically convex disks. Lastly, we prove a spherical variant of a theorem of Winternitz. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10614 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Steiner symmetrization on the sphere Basit, Bushra Hoehner, Steven Lángi, Zsolt Ledford, Jeff Metric Geometry The aim of this paper is to introduce a generalization of Steiner symmetrization in Euclidean space for spherical space, which is the dual of the Steiner symmetrization in hyperbolic space introduced by J. Schneider (Manuscripta Math. 60: 437-461, 1988). We show that this symmetrization preserves volume in every dimension, and convexity in the spherical plane, but not in dimensions $n > 2$. In addition, we investigate the monotonicity properties of the perimeter and diameter of a set under this process, and find conditions under which the image of a spherically convex disk under a suitable sequence of Steiner symmetrizations converges to a spherical cap. We apply our results to prove a spherical analogue of a theorem of Sas, and to confirm a conjecture of Besau and Werner (Adv. Math. 301: 867-901, 2016) for centrally symmetric spherically convex disks. Lastly, we prove a spherical variant of a theorem of Winternitz. |
| title | Steiner symmetrization on the sphere |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2406.10614 |