About: Simple Proofs of Classical Theorems in Discrete Geometry via the Guth-Katz Polynomial Partitioning Technique     Goto   Sponge   NotDistinct   Permalink

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  • Recently Guth and Katz (arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of ErdAs's distinct distances problem, a new method for partitioning finite point sets in R-d , based on the Stone-Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemer,di-Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach-Sharir bound on the number of incidences with algebraic curves of bounded degree.
  • Recently Guth and Katz (arXiv:1011.4105, 2010) invented, as a step in their nearly complete solution of ErdAs's distinct distances problem, a new method for partitioning finite point sets in R-d , based on the Stone-Tukey polynomial ham-sandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemer,di-Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at self-contained, expository treatment. We also mention some generalizations and extensions, such as the Pach-Sharir bound on the number of incidences with algebraic curves of bounded degree. (en)
Title
  • Simple Proofs of Classical Theorems in Discrete Geometry via the Guth-Katz Polynomial Partitioning Technique
  • Simple Proofs of Classical Theorems in Discrete Geometry via the Guth-Katz Polynomial Partitioning Technique (en)
skos:prefLabel
  • Simple Proofs of Classical Theorems in Discrete Geometry via the Guth-Katz Polynomial Partitioning Technique
  • Simple Proofs of Classical Theorems in Discrete Geometry via the Guth-Katz Polynomial Partitioning Technique (en)
skos:notation
  • RIV/00216208:11320/12:10125727!RIV13-MSM-11320___
http://linked.open...avai/riv/aktivita
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  • 167898
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  • RIV/00216208:11320/12:10125727
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  • Partitioning polynomial; Polynomial ham-sandwich; Spanning tree with low crossing number; Crossing number; Algebraic techniques; Incidences; PLANE; CURVES; APPROXIMATION; TREES; BOUNDS; VC-DIMENSION (en)
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  • DE - Spolková republika Německo
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  • [6D4F17DC3FC2]
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  • Discrete and Computational Geometry
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  • 48
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  • Matoušek, Jiří
  • Sharir, Micha
  • Kaplan, Haim
http://linked.open...ain/vavai/riv/wos
  • 000307507800001
issn
  • 0179-5376
number of pages
http://bibframe.org/vocab/doi
  • 10.1007/s00454-012-9443-3
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  • 11320
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