Width of convex bodies in hyperbolic space
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2023
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866916136697724928 |
|---|---|
| author | Lassak, Marek |
| author_facet | Lassak, Marek |
| contents | For every 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 $C$. We prove that if $\width_H (C) = Δ(C)$ and if there exists a unique most distant point $j \in C$ from $H$, then the projection of $j$ onto $H$ belongs to $H \cap C$. We verify that the diameter of $C$ equals to the maximum width of $C$. We define bodies of constant width in $\mathbb{H}^d$ in the standard way as bodies whose all widths are equal. We show that every body of constant width is strictly convex. 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 show that regular tetrahedra in $\mathbb{H}^3$ are not reduced. Similarly as in the Euclidean and spherical spaces, we introduce complete bodies and bodies of constant diameter in $\mathbb{H}^d$. We show that every body of constant width $δ$ is a body of constant diameter $δ$ and a complete body of diameter $δ$. Moreover, the two last conditions are equivalent. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_04412 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Width of convex bodies in hyperbolic space Lassak, Marek Metric Geometry 52A55 A.0 For every 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 $C$. We prove that if $\width_H (C) = Δ(C)$ and if there exists a unique most distant point $j \in C$ from $H$, then the projection of $j$ onto $H$ belongs to $H \cap C$. We verify that the diameter of $C$ equals to the maximum width of $C$. We define bodies of constant width in $\mathbb{H}^d$ in the standard way as bodies whose all widths are equal. We show that every body of constant width is strictly convex. 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 show that regular tetrahedra in $\mathbb{H}^3$ are not reduced. Similarly as in the Euclidean and spherical spaces, we introduce complete bodies and bodies of constant diameter in $\mathbb{H}^d$. We show that every body of constant width $δ$ is a body of constant diameter $δ$ and a complete body of diameter $δ$. Moreover, the two last conditions are equivalent. |
| title | Width of convex bodies in hyperbolic space |
| topic | Metric Geometry 52A55 A.0 |
| url | https://arxiv.org/abs/2306.04412 |