Roudneff's Conjecture in Dimension \(4\)

J.-P. Roudneff conjectured in 1991 that every arrangement of \(n \ge 2d+1\ge 5\) pseudohyperplanes in the real projective space \(\mathbb{P}^d\) has at most \(\sum_{i=0}^{d-2} \binom{n-1}{i}\) complete cells (i.e., cells bounded by each hyperplane). The conjecture is true for \(d=2,3\) and for arran...

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Bibliographic Details
Published inarXiv.org
Main Authors Hernández-Ortiz, Rangel, Knauer, Kolja, Luis Pedro Montejano, Scheucher, Manfred
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 24.03.2023
Subjects
Hyperplanes
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Summary:J.-P. Roudneff conjectured in 1991 that every arrangement of \(n \ge 2d+1\ge 5\) pseudohyperplanes in the real projective space \(\mathbb{P}^d\) has at most \(\sum_{i=0}^{d-2} \binom{n-1}{i}\) complete cells (i.e., cells bounded by each hyperplane). The conjecture is true for \(d=2,3\) and for arrangements arising from Lawrence oriented matroids. The main result of this manuscript is to show the validity of Roudneff's conjecture for \(d=4\). Moreover, based on computational data we conjecture that the maximum number of complete cells is only obtained by cyclic arrangements.
ISSN:2331-8422