article

Stabbing Simplices by Points and Flats

Authors:

Jiří Matoušek,

Gabriel NivaschAuthors Info & Claims

Discrete & Computational Geometry, Volume 43, Issue 2

Pages 321 - 338

Published: 01 March 2010 Publication History

Abstract

The following result was proved by Bárány in 1982: For every dź1, there exists cd>0 such that for every n-point set S in źd, there is a point pźźd contained in at least cdnd+1źO(nd) of the d-dimensional simplices spanned by S.

We investigate the largest possible value of cd. It was known that cd≤1/(2d(d+1)!) (this estimate actually holds for every point set S). We construct sets showing that cd≤(d+1)ź(d+1), and we conjecture that this estimate is tight. The best known lower bound, due to Wagner, is cdźźd:=(d2+1)/((d+1)!(d+1)d+1); in his method, p can be chosen as any centerpoint of S. We construct n-point sets with a centerpoint that is contained in no more than źdnd+1+O(nd) simplices spanned by S, thus showing that the approach using an arbitrary centerpoint cannot be further improved.

We also prove that for every n-point set Sźźd, there exists a (dź2)-flat that stabs at least cd,dź2n3źO(n2) of the triangles spanned by S, with $c_{d,d-2}\ge\frac{1}{24}(1-1/(2d-1)^{2})$ . To this end, we establish an equipartition result of independent interest (generalizing planar results of Buck and Buck and of Ceder): Every mass distribution in źd can be divided into 4dź2 equal parts by 2dź1 hyperplanes intersecting in a common (dź2)-flat.

References

[1]

Alon, N., Bárány, I., Füredi, Z., Kleitman, D.J.: Point selections and weak ¿-nets for convex hulls. Comb. Probab. Comput. 1, 189---200 (1992). http://www.math.tau.ac.il/~nogaa/PDFS/abfk3.pdf

[2]

Bárány, I.: A generalization of Carathéodory's theorem. Discrete Math. 40, 141---152 (1982)

Digital Library

[3]

Bárány, I., Füredi, Z., Lovász, L.: On the number of halving planes. Combinatorica 10(2), 175---183 (1990). http://www.cs.elte.hu/~lovasz/morepapers/halvingplane.pdf

[4]

Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry, 2nd edn. Algorithms and Computation in Mathematics, vol. 10. Springer, Berlin (2006)

Digital Library

[5]

Buck, R.C., Buck, E.F.: Equipartition of convex sets. Math. Mag. 22, 195---198 (1948---1949)

[6]

Boros, E., Füredi, Z.: Su un teorema di Kárteszi nella geometria combinatoria. Archimede 2, 71---76 (1977)

[7]

Boros, E., Füredi, Z.: The number of triangles covering the center of an n-set. Geom. Dedic. 17, 69---77 (1984)

[8]

Bukh, B.: A point in many triangles. Electron. J. Combin. 13(1), Note 10, 3 pp. (2006), (electronic)

[9]

Ceder, J.G.: Generalized sixpartite problems. Bol. Soc. Mat. Mex. (2) 9, 28---32 (1964)

[10]

Dey, T.K., Edelsbrunner, H.: Counting triangle crossings and halving planes. Discrete Comput. Geom. 12(1), 281---289 (1994)

Digital Library

[11]

Eppstein, D.: Improved bounds for intersecting triangles and halving planes. J. Comb. Theory Ser. A 62(1), 176---182 (1993). http://www.ics.uci.edu/~eppstein/pubs/Epp-TR-91-60.pdf

Digital Library

[12]

Fadell, E.R.: Ideal-valued generalizations of Ljusternik---Schnierelmann category, with applications. In: Fadell, E., et al. (eds.) Topics in Equivariant Topology. Sém. Math. Sup., vol. 108, pp. 11---54. Press Univ. Montréal, Montréal (1989)

[13]

Fadell, E.R., Husseini, S.: An ideal-valued cohomological index theory with applications to Borsuk---Ulam and Bourgin---Yang theorems. Ergod. Theory Dynam. Sys. 8, 73---85 (1988)

[14]

Inoue, A.: Borsuk-Ulam type theorems on Stiefel manifolds. Osaka J. Math. 43(1), 183---191 (2006). http://projecteuclid.org/euclid.ojm/1146243001

[15]

Kárteszi, F.: Extremalaufgaben über endliche Punktsysteme. Publ. Math. (Debr.) 4, 16---27 (1955)

[16]

Matoušek, J.: Lectures on Discrete Geometry. Springer, New York (2002)

[17]

Matoušek, J.: Using the Borsuk---Ulam Theorem. Springer, Berlin (2008). Revised 2nd printing

[18]

Moon, J.W.: Topics on Tournaments. Holt, Rinehart and Winston, New York (1968)

[19]

Nivasch, G., Sharir, M.: Eppstein's bound on intersecting triangles revisited. J. Comb. Theory Ser. A 116(2), 494---497 (2009). arXiv:0804.4415

Digital Library

[20]

Rado, R.: A theorem on general measure. J. Lond. Math. Soc. 21, 291---300 (1947)

[21]

Smorodinsky, S.: Combinatorial problems in computational geometry. Ph.D. thesis, Tel Aviv University, June 2003. http://www.cs.bgu.ac.il/~shakhar/my_papers/phd.ps.gz

[22]

Wagner, U.: On k-sets and applications. Ph.D. thesis, ETH Zürich, June 2003. http://www.inf.ethz.ch/~emo/DoctThesisFiles/wagner03.pdf

[23]

Welzl, E.: Entering and leaving j-facets. Discrete Comput. Geom. 25, 351---364 (2001). http://www.inf.ethz.ch/personal/emo/PublFiles/EnterLeave_DCG25_01.ps

Digital Library

[24]

Wendel, J.G.: A problem in geometric probability. Math. Scand. 11, 109---111 (1962)

[25]

¿ivaljević, R.T.: User's guide to equivariant methods in combinatorics II. Publ. Inst. Math. (Beogr.) 64(78), 107---132 (1998)

[26]

¿ivaljević, R.T.: Topological methods. In: Goodman, J.E., O'Rourke, J. (eds.) Handbook of Discrete and Computational Geometry, 2nd edn. CRC Press, Boca Raton (2004). Chap. 14

[27]

¿ivaljević, R.T., Vrećica, S.T.: The colored Tverberg's problem and complexes of injective functions. J. Comb. Theory Ser. A 61(2), 309---318 (1992)

Digital Library

Cited By

Schnider P(2020)Ham-Sandwich Cuts and Center Transversals in SubspacesDiscrete & Computational Geometry10.1007/s00454-020-00196-x64:4(1192-1209)Online publication date: 1-Dec-2020
https://dl.acm.org/doi/10.1007/s00454-020-00196-x
Blagojević PKarasev RMagazinov A(2018)A Center Transversal Theorem for an Improved Rado DepthDiscrete & Computational Geometry10.1007/s00454-018-0006-060:2(406-419)Online publication date: 1-Sep-2018
https://dl.acm.org/doi/10.1007/s00454-018-0006-0
Magazinov APór A(2018)An Improvement on the Rado Bound for the Centerline DepthDiscrete & Computational Geometry10.1007/s00454-016-9848-559:2(477-505)Online publication date: 1-Mar-2018
https://dl.acm.org/doi/10.1007/s00454-016-9848-5
Show More Cited By

Stabbing Simplices by Points and Flats
1. Theory of computation
  1. Randomness, geometry and discrete structures

Recommendations

Hitting Simplices with Points in ź3

The so-called first selection lemma states the following: given any set P of n points in źd, there exists a point in źd contained in at least cdnd+1źO(nd) simplices spanned by P, where the constant cd depends on d. We present improved bounds on the ...
Hitting Simplices with Points in ℝ³

The so-called first selection lemma states the following: given any set P of n points in ℝ^d, there exists a point in ℝ^d contained in at least c _d n ^d+1−O(n ^d) simplices spanned by P, where the constant c _d depends on d. We present improved bounds ...
Eppstein's bound on intersecting triangles revisited

Let S be a set of n points in the plane, and let T be a set of m triangles with vertices in S. Then there exists a point in the plane contained in @W(m^3/(n^6log^2n)) triangles of T. Eppstein [D. Eppstein, Improved bounds for intersecting triangles and ...

Comments

Information & Contributors

Information

Published In

cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 43, Issue 2

March 2010

295 pages

ISSN:0179-5376

Issue’s Table of Contents

Copyright © Copyright © 2010 Springer Science+Business Media, LLC.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 March 2010

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

11
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 10 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Schnider P(2020)Ham-Sandwich Cuts and Center Transversals in SubspacesDiscrete & Computational Geometry10.1007/s00454-020-00196-x64:4(1192-1209)Online publication date: 1-Dec-2020
https://dl.acm.org/doi/10.1007/s00454-020-00196-x
Blagojević PKarasev RMagazinov A(2018)A Center Transversal Theorem for an Improved Rado DepthDiscrete & Computational Geometry10.1007/s00454-018-0006-060:2(406-419)Online publication date: 1-Sep-2018
https://dl.acm.org/doi/10.1007/s00454-018-0006-0
Magazinov APór A(2018)An Improvement on the Rado Bound for the Centerline DepthDiscrete & Computational Geometry10.1007/s00454-016-9848-559:2(477-505)Online publication date: 1-Mar-2018
https://dl.acm.org/doi/10.1007/s00454-016-9848-5
Yehudayoff A(2017)An Elementary Exposition of Topological Overlap in the PlaneDiscrete & Computational Geometry10.1007/s00454-017-9910-y58:2(255-264)Online publication date: 1-Sep-2017
https://dl.acm.org/doi/10.1007/s00454-017-9910-y
Fabila-Monroy RHuemer C(2017)Carathéodory's Theorem in DepthDiscrete & Computational Geometry10.1007/s00454-017-9893-858:1(51-66)Online publication date: 1-Jul-2017
https://dl.acm.org/doi/10.1007/s00454-017-9893-8
Matoušek JWagner U(2014)On Gromov's Method of Selecting Heavily Covered PointsDiscrete & Computational Geometry10.1007/s00454-014-9584-752:1(1-33)Online publication date: 1-Jul-2014
https://dl.acm.org/doi/10.1007/s00454-014-9584-7
Král' DMach LSereni J(2012)A New Lower Bound Based on Gromov's Method of Selecting Heavily Covered PointsDiscrete & Computational Geometry10.5555/3116657.311697048:2(487-498)Online publication date: 1-Sep-2012
https://dl.acm.org/doi/10.5555/3116657.3116970
Fox JGromov MLafforgue VNaor APach JRandall D(2011)Overlap properties of geometric expandersProceedings of the twenty-second annual ACM-SIAM symposium on Discrete algorithms10.5555/2133036.2133126(1188-1197)Online publication date: 23-Jan-2011
https://dl.acm.org/doi/10.5555/2133036.2133126
Basit AMustafa NRay SRaza S(2010)Hitting Simplices with Points in ź3Discrete & Computational Geometry10.5555/3115968.311614544:3(637-644)Online publication date: 1-Oct-2010
https://dl.acm.org/doi/10.5555/3115968.3116145
Basit AMustafa NRay SRaza SKirkpatrick DMitchell J(2010)Improving the first selection lemma in R3Proceedings of the twenty-sixth annual symposium on Computational geometry10.1145/1810959.1811017(354-357)Online publication date: 13-Jun-2010
https://dl.acm.org/doi/10.1145/1810959.1811017
Show More Cited By

View Options

View options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents