R-hulloid of the vertices of a tetrahedron
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915759136964608 |
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| author | Longinetti, Marco Naldi, Simone Venturi, Adriana |
| author_facet | Longinetti, Marco Naldi, Simone Venturi, Adriana |
| contents | The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, of the set of vertices $V$ of a tetrahedron $T$ is the minimal closed set containing $V$ such that its complement is the union of open balls of radius $R$. When $R$ is greater than the circumradius of $T$, the boundary of the $R$-hulloid consists of $V$ and possibly of four spherical subsets of well defined spheres of radius $R$ through the vertices of $T$. The existence of a value $R^*$ such that these subsets collapse into a point $O^*$, in the interior of $T$, is investigated; in such a case $O^*$ belongs to four spheres of radius $R^*$, each one through three vertices of $T$ and not containing the fourth one. As a consequence, the range of $ρ$ such that $V$ is a $ρ$-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00658 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | R-hulloid of the vertices of a tetrahedron Longinetti, Marco Naldi, Simone Venturi, Adriana Metric Geometry 52A01, 52A30 The $R$-hulloid, in the Euclidean space $\mathbb{R}^3$, of the set of vertices $V$ of a tetrahedron $T$ is the minimal closed set containing $V$ such that its complement is the union of open balls of radius $R$. When $R$ is greater than the circumradius of $T$, the boundary of the $R$-hulloid consists of $V$ and possibly of four spherical subsets of well defined spheres of radius $R$ through the vertices of $T$. The existence of a value $R^*$ such that these subsets collapse into a point $O^*$, in the interior of $T$, is investigated; in such a case $O^*$ belongs to four spheres of radius $R^*$, each one through three vertices of $T$ and not containing the fourth one. As a consequence, the range of $ρ$ such that $V$ is a $ρ$-body is described completely. This work generalizes to dimension three previous results, proved in the planar case and related to the three circles Johnson's Theorem. |
| title | R-hulloid of the vertices of a tetrahedron |
| topic | Metric Geometry 52A01, 52A30 |
| url | https://arxiv.org/abs/2406.00658 |