Perpendicular Dissections of Space |
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Authors: | Thomas Zaslavsky |
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Institution: | (1) Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA zaslav@math.binghamton.edu, US |
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Abstract: | For each pair (Q
i
,Q
j
) of reference points and each real number r there is a unique hyperplane h \perp Q
i
Q
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.
Received July 20, 2000, and in revised form September 29, 2001, and October 12, 2001. Online publication March 4, 2002. |
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Keywords: | , 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 |
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