Pal's isominwidth problem in the hyperbolic space
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911172643520512 |
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| author | Boroczky, Karoly J. Freyer, Ansgar Sagmeister, Adam |
| author_facet | Boroczky, Karoly J. Freyer, Ansgar Sagmeister, Adam |
| contents | The paper focuses on possible hyperbolic versions of the classical Pal isominwidth inequality in R^2 from 1921, which states that for a fixed minimal width, the regular triangle has minimal area. We note that the isominwidth problem is still wide open in R^n for n>2. Recent work on the isominwidth problem on the sphere S^2 shows that the solution is the regular spherical triangle when the width is at most π/2 according to Bezdek and Blekherman, while Freyer and Sagmeister proved that the minimizer is the polar of a spherical Reuleaux triangle when the minimal width is greater than π/2.
In this paper, the hyperbolic isominwidth problem is discussed with respect to the probably most natural notion of width due to Lassak in the hyperbolic space H^n where strips bounded by a supporting hyperplane and a corresponding hypersphere are considered. On the one hand, we show that the volume of a convex body of given minimal Lassak width w>0 in H^n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in H^n does not make sense. On the other hand, in the two-dimensional case, we prove that among horocyclically convex bodies of given Lassak width in H^2, the area is minimized by the regular horocyclic triangle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_04427 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pal's isominwidth problem in the hyperbolic space Boroczky, Karoly J. Freyer, Ansgar Sagmeister, Adam Metric Geometry The paper focuses on possible hyperbolic versions of the classical Pal isominwidth inequality in R^2 from 1921, which states that for a fixed minimal width, the regular triangle has minimal area. We note that the isominwidth problem is still wide open in R^n for n>2. Recent work on the isominwidth problem on the sphere S^2 shows that the solution is the regular spherical triangle when the width is at most π/2 according to Bezdek and Blekherman, while Freyer and Sagmeister proved that the minimizer is the polar of a spherical Reuleaux triangle when the minimal width is greater than π/2. In this paper, the hyperbolic isominwidth problem is discussed with respect to the probably most natural notion of width due to Lassak in the hyperbolic space H^n where strips bounded by a supporting hyperplane and a corresponding hypersphere are considered. On the one hand, we show that the volume of a convex body of given minimal Lassak width w>0 in H^n might be arbitrarily small; therefore, the isominwidth problem for convex bodies in H^n does not make sense. On the other hand, in the two-dimensional case, we prove that among horocyclically convex bodies of given Lassak width in H^2, the area is minimized by the regular horocyclic triangle. |
| title | Pal's isominwidth problem in the hyperbolic space |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2502.04427 |