Steiner symmetrization on the sphere
| Autor: | Basit, Bushra, Hoehner, Steven, Lángi, Zsolt, Ledford, Jeff |
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| Rok vydání: | 2024 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 Peyerimhoff (J. London Math. Soc. (2) 66: 753-768, 2002). 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. We also prove a spherical variant of a theorem of Winternitz. Comment: 27 pages, 3 figures |
| Databáze: | arXiv |
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