Sphere-of-influence graphs in normed spaces

TL;DR

This paper proves that in any normed space, the maximum degree of the k-th sphere-of-influence graph is bounded by a function of the space's dimension and the parameter k, generalizing previous results.

Contribution

It establishes a universal degree bound for sphere-of-influence graphs in normed spaces, extending earlier specific case results.

Findings

Vertex of degree less than 5^d k in k-th sphere-of-influence graph

Generalization of previous bounds in Euclidean spaces to normed spaces

Unified bound applicable across different normed geometries

Abstract

We show that any $k$-th closed sphere-of-influence graph in a $d$-dimensional normed space has a vertex of degree less than~$5^d k$, thus obtaining a common generalization of results of F\"uredi and Loeb (1994) and Guibas, Pach and Sharir (1994).