Reduced polygons in the hyperbolic plane
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911907178348544 |
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| author | Lassak, Marek |
| author_facet | Lassak, Marek |
| contents | For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The minimum width of $C$ over all supporting $H$ is called the thickness $Δ(C)$ of $C$. A convex body $R \subset \mathbb{H}^d$ is said to be reduced if $Δ(Z) < Δ(R)$ for every convex body $Z$ properly contained in $R$. We describe a class of reduced polygons in $\mathbb{H}^2$ and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07831 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Reduced polygons in the hyperbolic plane Lassak, Marek Metric Geometry 52A55 For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The minimum width of $C$ over all supporting $H$ is called the thickness $Δ(C)$ of $C$. A convex body $R \subset \mathbb{H}^d$ is said to be reduced if $Δ(Z) < Δ(R)$ for every convex body $Z$ properly contained in $R$. We describe a class of reduced polygons in $\mathbb{H}^2$ and present some properties of them. In particular, we estimate their diameters in terms of their thicknesses. |
| title | Reduced polygons in the hyperbolic plane |
| topic | Metric Geometry 52A55 |
| url | https://arxiv.org/abs/2401.07831 |