On Spheres with $k$ Points Inside

TL;DR

This paper extends Delaunay triangulation concepts to spheres with exactly k points inside, providing coverage and combinatorial results for weighted and finite point sets, with applications to hyperplane arrangements and hypersimplex volumes.

Contribution

It generalizes the classic Delaunay triangulation to include spheres with k points inside and proves new coverage and combinatorial properties for such simplices.

Findings

Simplices with k points inside cover space exactly inom{d+k}{d} times.

The subset of these simplices incident to a point covers neighborhoods inom{d+k-1}{d-1} times.

New proofs for classic results on k-facets, hyperplane arrangements, and hypersimplex volumes.

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.