The Dirac–Goodman–Pollack Conjecture.
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| Title: | The Dirac–Goodman–Pollack Conjecture. |
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| Authors: | Dumitrescu, Adrian1 (AUTHOR) ad.dumitrescu@algoresearch.org |
| Source: | Discrete & Computational Geometry. Sep2024, Vol. 72 Issue 2, p728-742. 15p. |
| Subjects: | Discrete geometry, Logical prediction, Generalization, Permutations |
| Abstract: | In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of n noncollinear points in the plane has a point incident to at least cn connecting lines determined by the set. The notion of allowable sequences of permutations provides a natural combinatorial setting for analyzing these problems. Within this formalism, the conjectured generalization reads as follows: Any nontrivial allowablen-sequence Σ has a local sequence Λ i whose half-period is at leastcn. The conjecture is confirmed here with a concrete bound c = 1 / 845 . Several related problems are discussed. [ABSTRACT FROM AUTHOR] |
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| Database: | Engineering Source |
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| Abstract: | In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of n noncollinear points in the plane has a point incident to at least cn connecting lines determined by the set. The notion of allowable sequences of permutations provides a natural combinatorial setting for analyzing these problems. Within this formalism, the conjectured generalization reads as follows: Any nontrivial allowablen-sequence Σ has a local sequence Λ i whose half-period is at leastcn. The conjecture is confirmed here with a concrete bound c = 1 / 845 . Several related problems are discussed. [ABSTRACT FROM AUTHOR] |
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| ISSN: | 01795376 |
| DOI: | 10.1007/s00454-023-00487-z |
