On Spheres with $k$ Points Inside

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.