Name
Abstract | ||
|---|---|---|
For each pair (Qi,Qj) of reference points and each real number r there is a unique hyperplane h \\perp QiQj such that d(P,Qi)2 - d(P,Qj)2 = r for points P in h . Take n reference points in d -space and for each pair (Qi,Qj) 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. |
Year | DOI | Venue |
|---|---|---|
2002 | 10.1007/s00454-001-0073-4 | Discrete & Computational Geometry |
Keywords | Field | DocType |
Arrangement of hyperplanes,affinographic arrangement,deformation of Coxeter arrangement,additive real gain graph,graphic lift matroid,concurrence of perpendiculars,Pythagorean theorem,perpendicular bisector,intersection semilattice,geometric semilattice,balanced chromatic polynomial,Whitney numbers,composed partition,fat forest | Topology,Discrete mathematics,Combinatorics,Gain graph,Finite set,Arrangement of hyperplanes,Infinity,Semilattice,Hyperplane,Pythagorean theorem,Real number,Mathematics | Journal |
Volume | Issue | ISSN |
27 | 3 | Discrete and Computational Geometry, 27 (2002), 303--351 |
Citations | PageRank | References |
4 | 0.62 | 6 |
Authors | ||
1 | ||
Name | Order | Citations | PageRank |
|---|---|---|---|
| T. Zaslavsky | 1 | 297 | 56.67 |