Perpendicular dissections of space

TL;DR

This paper studies the geometric and combinatorial structure of hyperplane arrangements formed by perpendicular bisectors and related constructs in high-dimensional space, linking them to gain graphs and exploring their properties.

Contribution

It introduces a novel connection between perpendicular hyperplane arrangements and real, additive gain graphs, and analyzes their intersection semilattices and combinatorial properties.

Findings

The intersection semilattice corresponds to the structure of balanced flats in a gain graph.

Identifies conditions for genericity related to behavior at infinity.

Provides examples including arrangements of perpendicular bisectors and their combinatorial counts.

Abstract

For each pair $(Q_i,Q_j)$ of reference points and each real number $r$ there is a unique hyperplane $h \perp Q_iQ_j$ such that $d(P,Q_i)^2 - d(P,Q_j)^2 = r$ for points $P$ in $h$. Take $n$ reference points in $d$-space and for each pair $(Q_i,Q_j)$ a finite set of real numbers. The corresponding perpendiculars form an arrangement of hyperplanes. We explore the structure of the semilattice of intersections of the hyperplanes for generic reference points. The main theorem is that there is a real, additive gain graph (this is a graph with an additive real number associated invertibly to each edge) whose set of balanced flats has the same structure as the intersection semilattice. We examine the requirements for genericity, which are related to behavior at infinity but remain mysterious; also, variations in the construction rules for perpendiculars. We investigate several particular arrangements with a view to finding the exact numbers of faces of each dimension. The prototype, the arrangement of all perpendicular bisectors, was studied by Good and Tideman, motivated by a geometric voting theory. Most of our particular examples are suggested by extensions of that theory in which voters exercise finer discrimination. Throughout, we propose many research problems.