On Spheres with $k$ Points Inside

Herbert Edelsbrunner; Alexey Garber; Morteza Saghafian

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arXiv · 2410.21204

On Spheres with $k$ Points Inside

Herbert Edelsbrunner·Alexey Garber·Morteza Saghafian

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Abstract

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.

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Herbert Edelsbrunner, Alexey Garber, Morteza Saghafian. 2024-10-28. On Spheres with $k$ Points Inside. https://arxiv.org/abs/2410.21204

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