, Volume 38, Issue 5, pp 1149–1174 | Cite as

Essential Dimension and the Flats Spanned by a Point Set

  • Ben LundEmail author
Original Paper


Let P be a finite set of points in ℝd or ℂd.We answer a question of Purdy on the conditions under which the number of hyperplanes spanned by P is at least the number of (d−2)-flats spanned by P.

In answering this question, we define a new measure of the degeneracy of a point set with respect to affine subspaces, termed the essential dimension. We use the essential dimension to give an asymptotic expression for the number of k-flats spanned by P, for 1≤kd−1.

Mathematics Subject Classification (2000)



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  1. [1]
    O. Aichholzer and F. Aurenhammer: Classifying hyperplanes in hypercubes, SIAM Journal on Discrete Mathematics 9 (1996), 225–232.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    G. L. Alexanderson and J. E. Wetzel: A simplicial 3-arrangement of 21 planes, Discrete mathematics 60 (1986), 67–73.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    J. Beck: On the lattice property of the plane and some problems of Dirac, Motzkin and Erdős in combinatorial geometry, Combinatorica 3 (1983), 281–297.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    P. Brass, W. O. J. Moser and J. Pach: Research problems in discrete geometry, Springer Science & Business Media, 2005.zbMATHGoogle Scholar
  5. [5]
    H. T. Croft, K. J. Falconer and R. K. Guy: Unsolved Problems in Geometry: Unsolved Problems in Intuitive Mathematics, volume 2, Springer Science & Business Media, 2012.zbMATHGoogle Scholar
  6. [6]
    N. G. de Bruijn and P. Erdős: On a combinatorial problem, Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen Indagationes mathematicae 51 (1948), 1277–1279.zbMATHGoogle Scholar
  7. [7]
    T. Do: Extending Erdős-Beck’s theorem to higher dimensions, arXiv preprint arXiv:1607.00048, 2016.Google Scholar
  8. [8]
    P. Erdős and George Purdy: Extremal problems in combinatorial geometry, in: Handbook of combinatorics (vol. 1), 809–874. MIT Press, 1996.Google Scholar
  9. [9]
    R. Gian-Carlo: Combinatorial theory, old and new, in: Proceedings of the International Mathematical Congress Held..., volume 3, 229, University of Toronto Press, 1971.Google Scholar
  10. [10]
    B. Grünbaum: A catalogue of simplicial arrangements in the real projective plane, Ars Mathematica Contemporanea 2 (2009).Google Scholar
  11. [11]
    B. Grünbaum and G. C. Shephard: Simplicial arrangements in projective 3-space, Mitt. Math. Semin. Giessen 166 (1984), 49–101.MathSciNetzbMATHGoogle Scholar
  12. [12]
    B. D. Lund, G. B Purdy and J. W. Smith: A bichromatic incidence bound and an application, Discrete & Computational Geometry 46 (2011), 611–625.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    J. H. Mason: Matroids: Unimodal conjectures and Motzkins theorem, Combinatorics (D. JA Welsh and DR Woodall, eds.), Institute of Math. and Appl, 207–221, 1972.Google Scholar
  14. [14]
    G. Purdy: Two results about points, lines and planes, Discrete mathematics 60 (1986), 215–218.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    E. Szemerédi and W. T. Trotter Jr: Extremal problems in discrete geometry, Combinatorica 3 (1983), 381–392.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    Cs. D Tóth: The Szemerédi-Trotter theorem in the complex plane, Combinatorica 35 (2015), 95–126.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    J. Zahl: A Szemerédi-Trotter type theorem in ℝ4, Discrete & Computational Geometry 54 (2015), 513–572.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of GeorgiaAthensUSA

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