Applications of equidistant supporting surfaces of a convex body in the hyperbolic space
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917663626756096 |
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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 thickness (i.e., the minimum width) of $C$ is denoted by $Δ(C)$. A convex body $R \subset \mathbb{H}^d$ is called reduced if for every body $Z \subsetneq R$ we have $Δ(Z) < Δ(R)$. We show that for any extreme point $e$ of a reduced body $R \subset \mathbb{H}^d$ there exists a supporting hyperplane $H$ of $R$ which passes through $e$ or its equidistant surface supporting $R$ passes through $e$. Bodies of constant width in $\mathbb{H}^d$ are defined as bodies whose all widths are equal. We prove that every complete body in $\mathbb{H}^d$ is a body of constant width. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_07900 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Applications of equidistant supporting surfaces of a convex body in the hyperbolic space 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 thickness (i.e., the minimum width) of $C$ is denoted by $Δ(C)$. A convex body $R \subset \mathbb{H}^d$ is called reduced if for every body $Z \subsetneq R$ we have $Δ(Z) < Δ(R)$. We show that for any extreme point $e$ of a reduced body $R \subset \mathbb{H}^d$ there exists a supporting hyperplane $H$ of $R$ which passes through $e$ or its equidistant surface supporting $R$ passes through $e$. Bodies of constant width in $\mathbb{H}^d$ are defined as bodies whose all widths are equal. We prove that every complete body in $\mathbb{H}^d$ is a body of constant width. |
| title | Applications of equidistant supporting surfaces of a convex body in the hyperbolic space |
| topic | Metric Geometry 52A55 |
| url | https://arxiv.org/abs/2403.07900 |