On Spheres with $k$ Points Inside
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916935103414272 |
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| author | Edelsbrunner, Herbert Garber, Alexey Saghafian, Morteza |
| author_facet | Edelsbrunner, Herbert Garber, Alexey Saghafian, Morteza |
| contents | We generalize the classic definition of Delaunay triangulation and prove that for a locally finite and coarsely dense generic point set, $A \subseteq \mathbb{R}^d$, the $d$-simplices whose vertices belong to $A$ and whose circumscribed spheres enclose exactly $k$ points of $A$ cover $\mathbb{R}^d$ exactly $\binom{d+k}{d}$ times. Similarly, the subset of such simplices incident to a point in $A$ cover any small enough neighborhood of that point exactly $\binom{d+k-1}{d-1}$ times. We extend this result to the cases in which the points are weighted and when $A$ contains only finitely many points in $\mathbb{R}^d$ or in $\mathbb{S}^d$. Using these results, we give new proofs of classic results on $k$-facets, old and new combinatorial results for hyperplane arrangements, and a new proof for the fact that the volumes of hypersimplices are Eulerian numbers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_21204 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Spheres with $k$ Points Inside Edelsbrunner, Herbert Garber, Alexey Saghafian, Morteza Combinatorics Computational Geometry We generalize the classic definition of Delaunay triangulation and prove that for a locally finite and coarsely dense generic point set, $A \subseteq \mathbb{R}^d$, the $d$-simplices whose vertices belong to $A$ and whose circumscribed spheres enclose exactly $k$ points of $A$ cover $\mathbb{R}^d$ exactly $\binom{d+k}{d}$ times. Similarly, the subset of such simplices incident to a point in $A$ cover any small enough neighborhood of that point exactly $\binom{d+k-1}{d-1}$ times. We extend this result to the cases in which the points are weighted and when $A$ contains only finitely many points in $\mathbb{R}^d$ or in $\mathbb{S}^d$. Using these results, we give new proofs of classic results on $k$-facets, old and new combinatorial results for hyperplane arrangements, and a new proof for the fact that the volumes of hypersimplices are Eulerian numbers. |
| title | On Spheres with $k$ Points Inside |
| topic | Combinatorics Computational Geometry |
| url | https://arxiv.org/abs/2410.21204 |