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A new lower bound based on Gromov’s method of selecting heavily covered points
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Král’, Daniel, Mach, Lukáš and Sereni, Jean-Sébastien (2012) A new lower bound based on Gromov’s method of selecting heavily covered points. Discrete & Computational Geometry, Vol.48 (No.2). pp. 487-498. doi:10.1007/s00454-012-9419-3 ISSN 0179-5376.
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Official URL: http://dx.doi.org/10.1007/s00454-012-9419-3
Abstract
Boros and Füredi (for d=2) and Bárány (for arbitrary d) proved that there exists a positive real number c d such that for every set P of n points in R d in general position, there exists a point of R d contained in at least cd(nd+1) d-simplices with vertices at the points of P. Gromov improved the known lower bound on c d by topological means. Using methods from extremal combinatorics, we improve one of the quantities appearing in Gromov’s approach and thereby provide a new stronger lower bound on c d for arbitrary d. In particular, we improve the lower bound on c 3 from 0.06332 to more than 0.07480; the best upper bound known on c 3 being 0.09375.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Discrete & Computational Geometry | ||||
Publisher: | Springer New York LLC | ||||
ISSN: | 0179-5376 | ||||
Official Date: | 2012 | ||||
Dates: |
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Volume: | Vol.48 | ||||
Number: | No.2 | ||||
Page Range: | pp. 487-498 | ||||
DOI: | 10.1007/s00454-012-9419-3 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published |
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