Essential Dimension and the Flats Spanned by a Point Set

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    O. Aichholzer and F. Aurenhammer: Classifying hyperplanes in hypercubes, SIAM Journal on Discrete Mathematics 9 (1996), 225–232.

    MathSciNet  Article  MATH  Google Scholar 

  2. [2]

    G. L. Alexanderson and J. E. Wetzel: A simplicial 3-arrangement of 21 planes, Discrete mathematics 60 (1986), 67–73.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  4. [4]

    P. Brass, W. O. J. Moser and J. Pach: Research problems in discrete geometry, Springer Science & Business Media, 2005.

    MATH  Google 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.

    MATH  Google 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.

    MATH  Google 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.

    MathSciNet  MATH  Google 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.

    MathSciNet  Article  MATH  Google 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.

    MathSciNet  Article  MATH  Google Scholar 

  15. [15]

    E. Szemerédi and W. T. Trotter Jr: Extremal problems in discrete geometry, Combinatorica 3 (1983), 381–392.

    MathSciNet  Article  MATH  Google Scholar 

  16. [16]

    Cs. D Tóth: The Szemerédi-Trotter theorem in the complex plane, Combinatorica 35 (2015), 95–126.

    MathSciNet  Article  MATH  Google Scholar 

  17. [17]

    J. Zahl: A Szemerédi-Trotter type theorem in ℝ4, Discrete & Computational Geometry 54 (2015), 513–572.

    MathSciNet  Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ben Lund.

Additional information

Work on this paper was supported by NSF grant CCF-1350572 and by ERC Grant 267165 DISCONV.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Lund, B. Essential Dimension and the Flats Spanned by a Point Set. Combinatorica 38, 1149–1174 (2018). https://doi.org/10.1007/s00493-016-3602-8

Download citation

Mathematics Subject Classification (2000)

  • 52C10