Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-23T14:15:02.420Z Has data issue: false hasContentIssue false
  • English
  • Français

Unit Distances in Three Dimensions

Published online by Cambridge University Press:  25 April 2012

HAIM KAPLAN ,
JIŘÍ MATOUŠEK ,
ZUZANA SAFERNOVÁ  and
MICHA SHARIR
HAIM KAPLAN
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: [email protected])
JIŘÍ MATOUŠEK
Affiliation:
Department of Applied Mathematics and Institute of Theoretical Computer Science (ITI), Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic (e-mail: [email protected]) and Institute of Theoretical Computer Science, ETH Zurich, 8092 Zurich, Switzerland
ZUZANA SAFERNOVÁ
Affiliation:
Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic (e-mail: [email protected])
MICHA SHARIR
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the number of unit distances determined by n points in ℝ3 is O(n3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].

Keywords

Primary 52C10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 4 , July 2012 , pp. 597 - 610
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Akama, Y., Irie, K., Kawamura, A. and Uwano, Y. (2010) VC dimensions of principal component analysis. Discrete Comput. Geom. 44 589–598.CrossRefGoogle Scholar
[2]Barone, S. and Basu, S. (2012) Refined bounds on the number of connected components of sign conditions on a variety. Discrete Comput. Geom. 47 (3)577–597.CrossRefGoogle Scholar
[3]Basu, S., Pollack, R. and Roy, M.-F. (1996) On the number of cells defined by a family of polynomials on a variety. Mathematika 43 120–126.CrossRefGoogle Scholar
[4]Basu, S., Pollack, R. and Roy, M.-F. (2003) Algorithms in Real Algebraic Geometry, Vol. 10 of Algorithms and Computation in Mathematics, Springer.CrossRefGoogle Scholar
[5]Clarkson, K., Edelsbrunner, H., Guibas, L., Sharir, M. and Welzl, E. (1990) Combinatorial complexity bounds for arrangements of curves and spheres. Discrete Comput. Geom. 5 99–160.CrossRefGoogle Scholar
[6]Cox, D., Little, J. and O'Shea, D. (2005) Using Algebraic Geometry, second edition, Springer.Google Scholar
[7]Cox, D., Little, J. and O'Shea, D. (2007) Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, third edition, Springer.CrossRefGoogle Scholar
[8]Elekes, G., Kaplan, H. and Sharir, M. (2011) On lines, joints, and incidences in three dimensions. J. Combin. Theory Ser. A 118 962–977. Also in arXiv:0905.1583.CrossRefGoogle Scholar
[9]Erdős, P. (1946) On a set of distances of n points. Amer. Math. Monthly 53 248–250.CrossRefGoogle Scholar
[10]Erdős, P. (1960) On sets of distances on n points in Euclidean space. Magyar Tud. Akad. Mat. Kutató Int. Kozl. 5 165–169.Google Scholar
[11]Guth, L. and Katz, N. H. (2010) Algebraic methods in discrete analogs of the Kakeya problem, Adv. Math. 225 2828–2839. Also in arXiv:0812.1043v1.CrossRefGoogle Scholar
[12]Guth, L. and Katz, N. H. (2010) On the Erdős distinct distances problem in the plane. arXiv:1011.4105.Google Scholar
[13]Harnack, C. G. A. (1876) Über die Vielfaltigkeit der ebenen algebraischen Kurven. Math. Ann. 10 189–199.CrossRefGoogle Scholar
[14]Kaplan, H., Matoušek, J. and Sharir, M. (2011) Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique. Discrete Comput. Geom., submitted. Also in arXiv:1102.5391.Google Scholar
[15]Lenz, H. (1955) Zur Zerlegung von Punktmengen in solche kleineren Durchmessers. Arch. Math. 6 413–416.CrossRefGoogle Scholar
[16]Matoušek, J. (2003) Using the Borsuk–Ulam Theorem, Springer.Google Scholar
[17]Milnor, J. (1964) On the Betti numbers of real varieties. Proc. Amer. Math. Soc. 15 275–280.CrossRefGoogle Scholar
[18]Oleinik, O. A. and Petrovskii, I. B. (1949) On the topology of real algebraic surfaces. Izv. Akad. Nauk SSSR 13 389–402.Google Scholar
[19]Pach, J. and Agarwal, P. K. (1995) Combinatorial Geometry, Wiley-Interscience.CrossRefGoogle Scholar
[20]Pach, J. and Sharir, M. (2004) Geometric incidences. In Towards a Theory of Geometric Graphs (Pach, J., ed.), Vol. 342 of Contemporary Mathematics, AMS, pp. 185–223.CrossRefGoogle Scholar
[21]Solymosi, J. and Tao, T. (2012) An incidence theorem in higher dimensions. arXiv:1103.2926v4CrossRefGoogle Scholar
[22]Spencer, J., Szemerédi, E. and Trotter, W. T. (1984) Unit distances in the Euclidean plane. In Graph Theory and Combinatorics: Proc. Cambridge Conf. on Combinatorics (Bollobás, B., ed.), Academic Press, pp. 293–308.Google Scholar
[23]Stone, A. H. and Tukey, J. W. (1942) Generalized sandwich theorems. Duke Math. J. 9 356–359.CrossRefGoogle Scholar
[24]Székely, L. (1997) Crossing numbers and hard Erdős problems in discrete geometry. Combinat. Probab. Comput. 6 353–358.CrossRefGoogle Scholar
[25]Thom, R. (1965) Sur l'homologie des variétés algébriques réelles. In Differential and Combinatorial Topology (Cairns, S. S., ed.), Princeton University Press, pp. 255–265.CrossRefGoogle Scholar
[26]Valtr, P. (2006) Strictly convex norms allowing many unit distances and related touching questions, manuscript.Google Scholar
[27]Warren, H. E. (1968) Lower bound for approximation by nonlinear manifolds. Trans. Amer. Math. Soc. 133 167–178.CrossRefGoogle Scholar
[28]Zahl, J. (2011) An improved bound on the number of point-surface incidences in three dimensions. arXiv:1104.4987. First posted (v1) 26 April 2011; revised and corrected 22 September 2011.Google Scholar