On Spheres with k Points Inside

2024-10-28

International Symposium on Computational Geometry

AI Summary

This paper generalizes Delaunay triangulations by introducing and proving an exact counting theorem for generalized *k*-empty-sphere *d*-simplices: for any locally finite, coarsely dense point set *A* in ℝᵈ, the number of *d*-simplices with vertices in *A* whose circumscribing sphere contains exactly *k* points of *A* is uniformly equal to the binomial coefficient *C(d + k, d)*. Methodologically, the proof integrates combinatorial geometry, weighted Voronoi analysis, spherical geometry, and topological tools, establishing for the first time the consistency of this count across Euclidean and spherical spaces, and extending it to weighted point sets and finite configurations. Key contributions include: (1) a unified combinatorial characterization of *k*-empty-sphere structures; (2) a purely geometric proof that the volume of the *d*-dimensional hypersimplex equals the Eulerian number *A(d, k)*; and (3) novel derivations of classical results—including *k*-face enumeration and hyperplane arrangement properties—via this geometric-combinatorial bridge.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We generalize a classical result by Boris Delaunay that introduced Delaunay triangulations. In particular, we prove that for a locally finite and coarsely dense generic point set $A$ in $mathbb{R}^d$, every generic point of $mathbb{R}^d$ belongs to exactly $inom{d+k}{d}$ simplices whose vertices belong to $A$ and whose circumspheres enclose exactly $k$ points of $A$. 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 $mathbb{S}^d$. Furthermore, we use the result to give a new geometric proof for the fact that volumes of hypersimplices are Eulerian numbers.

Problem

• Generalizing Delaunay triangulation for weighted point sets
• Proving coverage properties of k-point enclosing simplices
• Extending combinatorial results to hypersphere arrangements

Innovation

• Generalized Delaunay triangulation with k interior points
• Extended to weighted points and finite sets
• Applied to prove classic combinatorial geometry results