Approximation of spherical convex bodies of constant width $π/2$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913764846075904 |
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| author | Han, Huhe |
| author_facet | Han, Huhe |
| contents | Let $C\subset \mathbb{S}^2$ be a spherical convex body of constant width $τ$. It is known that (i) if $τ<π/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $τ$ whose boundary consists only of arcs of circles of radius $τ$ such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$; (ii) if $τ>π/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $τ$ whose boundary consists only of arcs of circles of radius $τ-\fracπ{2}$ and great circle arcs such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$. In this paper, we present an approximation of the remaining case $τ=π/2$, that is, if $τ=π/2$ then for any $\varepsilon>0$ there exists a spherical polytope $\mathcal{P}_\varepsilon$ of constant width $π/2$ such that the Hausdorff distance between $C$ and $\mathcal{P}_\varepsilon$ is at most $\varepsilon$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00596 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximation of spherical convex bodies of constant width $π/2$ Han, Huhe Metric Geometry Let $C\subset \mathbb{S}^2$ be a spherical convex body of constant width $τ$. It is known that (i) if $τ<π/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $τ$ whose boundary consists only of arcs of circles of radius $τ$ such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$; (ii) if $τ>π/2$ then for any $\varepsilon>0$ there exists a spherical convex body $C_\varepsilon$ of constant width $τ$ whose boundary consists only of arcs of circles of radius $τ-\fracπ{2}$ and great circle arcs such that the Hausdorff distance between $C$ and $C_\varepsilon$ is at most $\varepsilon$. In this paper, we present an approximation of the remaining case $τ=π/2$, that is, if $τ=π/2$ then for any $\varepsilon>0$ there exists a spherical polytope $\mathcal{P}_\varepsilon$ of constant width $π/2$ such that the Hausdorff distance between $C$ and $\mathcal{P}_\varepsilon$ is at most $\varepsilon$. |
| title | Approximation of spherical convex bodies of constant width $π/2$ |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2409.00596 |