article

Bounding the piercing number

Authors:

G. KalaiAuthors Info & Claims

Discrete & Computational Geometry, Volume 13, Issue 3-4

Pages 245 - 256

https://doi.org/10.1007/BF02574042

Published: 01 December 1995 Publication History

Abstract

It is shown that for everyk and everyp q d+1 there is ac=c(k,p,q,d)< such that the following holds. For every family whose members are unions of at mostk compact convex sets inRd in which any set ofp members of the family contains a subset of cardinalityq with a nonempty intersection there is a set of at mostc points inRd that intersects each member of . It is also shown that for everyp q d+1 there is aC=C(p,q,d)< such that, for every family[Figure not available: see fulltext.] of compact, convex sets inRd so that among andp of them someq have a common hyperplane transversal, there is a set of at mostC hyperplanes that together meet all the members of[Figure not available: see fulltext.].

References

[1]

N. Alon, I. Bárány, Z. Füredi, and D. J. Kleitman, Point selections and weak ¿-nets for convex hulls,Combin. Probab. Comput.1 (1992), 189---200.

[2]

N. Alon and G. Kalai, A simple proof of the upper bound theorem,European J. Combin.6 (1985), 211---214.

[3]

N. Alon and D. J. Kleitman, Piercing convex sets and the Hadwiger Debrunner (p, q)-problemAdv. in Math.96 (1992), 103---112. See also: Piercing convex sets (research announcement),Bull. Adv. in Math. Soc.27 (1992), 252---256.

[4]

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

Digital Library

[5]

S. Cappell, J. Goodman, J. Pach, R. Pollack, M. Sharir, and Rephael Wenger, Common tangents and common transversals,Adv. in Math.,106 (1994), 198---215.

[6]

B. Chazelle and J. Friedman, A deterministic view of random sampling and its use in geometry,Combinatorica10 (1990), 229---249.

[7]

L. Danzer, B. Grünbaum, and V. Klee,Helly's Theorem and Its Relatives, Proceedings of Symposium in Pure Mathematics, Vol. 7 (Convexity), American Mathematical Society, Providence, RI, 1963, pp. 101---180.

[8]

V. L. Dol'nikov, A coloring problem,Siberian Math. J13 (1982), 886---894; translated fromSibirsk. Mat. Zh.13 (1972), 1272---1283.

[9]

J. Eckhoff, An upper bound theorem for families of convex sets,Geom. Dedicata19 (1985), 217---227.

[10]

J. Eckhoff, Helly, Radon, and Carathèodory type theorems, inHandbook of Convex Geometry (P. Grüber and J. Wills, eds.), North-Holland, Amsterdam, 1993, pp. 389---448.

[11]

J. Eckhoff, A Gallai type transversal problem in the plane,Discrete Comput. Geom.9 (1993), 203---214.

Digital Library

[12]

B. E. Fullbright, Intersectional properties of certain families of compact convex sets,Pacific J. Math.50 (1974), 57---62.

[13]

B. Grünbaum, On intersections of similar sets,Portugal Math.,18 (1959), 155---164.

[14]

B. Grünbaum, Lectures on Combinatorial Geometry, Mimeographed Notes, University of Washington, Seattle, 1974.

[15]

B. Grünbaum and T. Motzkin, On components in some families of sets,Proc. Amer. Math. Soc.12 (1961), 607---613.

[16]

H. Hadwiger and H. Debrunner, Über eine Variante zum Helly'schen Satz,Arch. Math.8 (1957), 309---313.

[17]

H. Hadwiger, H. Debrunner, and V. Klee,Combinatorial Geometry in the Plane, Holt, Rinehart, and Winston, New York, 1964.

[18]

E. Helly, Über Mengen konvexer Körper mit gemeinschaftlichen Punkten,Jahresber. Deusch. Math.-Verein.32 (1923), 175---176.

[19]

G. Kalai, Intersection patterns of convex sets,Israel J. Math.48 (1984), 161---174.

[20]

M. Katchalski and A. Liu, A problem of geometry inRd,Proc. Amer. Math. Soc.75 (1979), 284---288.

[21]

H. Morris, Two pigeonhole principles and unions of convexly disjoint sets, Ph.D. thesis, California Institute of Technology, 1973.

[22]

A. Schrijver,Theory of Linear and Integer Programming, Wiley, New York, 1986.

[23]

H. Tverberg, A generalization of Radon's theorem.J. London Math. Soc.41 (1966), 123---128.

[24]

G. Wegner, Über eine kombinatorisch-geometrische Frage von Hadwiger und Debrunner,Israel J. Math.3 (1965), 187---198.

[25]

G. Wegner,d-collapsing and nerves of families of convex sets,Arch. Math.26 (1975), 317---321.

[26]

G. Wegner, Über Helly-Gallaische Stichzahlprobleme, 3,Koll. Discrete Geometry, Salzburg (Institut für Mathematik, Universität Salzburg), 1985, pp. 277---282.

Cited By

Rubin N(2022)An Improved Bound for Weak Epsilon-nets in the PlaneJournal of the ACM10.1145/355598569:5(1-35)Online publication date: 27-Oct-2022
https://dl.acm.org/doi/10.1145/3555985
Rubin NKhuller SVassilevska Williams V(2021)Stronger bounds for weak epsilon-nets in higher dimensionsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451062(989-1002)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451062
Martínez-Sandoval LRoldán-Pensado ERubin N(2020)Further Consequences of the Colorful Helly HypothesisDiscrete & Computational Geometry10.1007/s00454-019-00085-y63:4(848-866)Online publication date: 1-Jun-2020
https://dl.acm.org/doi/10.1007/s00454-019-00085-y
Show More Cited By

Index Terms

Bounding the piercing number
1. Mathematics of computing
  1. Discrete mathematics
2. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

Index terms have been assigned to the content through auto-classification.

Recommendations

Colorful Coverings of Polytopes and Piercing Numbers of Colorful d-Intervals
Abstract
We prove a common strengthening of Bárány’s colorful Carathéodory theorem and the KKMS theorem. In fact, our main result is a colorful polytopal KKMS theorem, which extends a colorful KKMS theorem due to Shih and Lee [Math. Ann. 296 (1993), no. 1, ...
Piercing Translates and Homothets of a Convex Body

According to a classical result of Gr nbaum, the transversal number ( ) of any family of pairwise-intersecting translates or homothets of a convex body C in ^d is bounded by a function of d . Denote by ( C ) (resp. ( C )) the supremum of the ...
About an Erdős---Grünbaum Conjecture Concerning Piercing of Non-bounded Convex Sets

In this paper, we study the number of compact sets needed in an infinite family of convex sets with a local intersection structure to imply a bound on its piercing number, answering a conjecture of Erd s and Grünbaum. Namely, if in an infinite family of ...

Comments

Information & Contributors

Information

Published In

cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 13, Issue 3-4

June 1995

382 pages

ISSN:0179-5376

Issue’s Table of Contents

Copyright © Copyright © 1995 Springer-Verlag New York Inc.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 1995

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 Aug 2024

Other Metrics

View Author Metrics

Citations

Cited By

Rubin N(2022)An Improved Bound for Weak Epsilon-nets in the PlaneJournal of the ACM10.1145/355598569:5(1-35)Online publication date: 27-Oct-2022
https://dl.acm.org/doi/10.1145/3555985
Rubin NKhuller SVassilevska Williams V(2021)Stronger bounds for weak epsilon-nets in higher dimensionsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451062(989-1002)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451062
Martínez-Sandoval LRoldán-Pensado ERubin N(2020)Further Consequences of the Colorful Helly HypothesisDiscrete & Computational Geometry10.1007/s00454-019-00085-y63:4(848-866)Online publication date: 1-Jun-2020
https://dl.acm.org/doi/10.1007/s00454-019-00085-y
Pinchasi R(2015)A Note on Smaller Fractional Helly NumbersDiscrete & Computational Geometry10.1007/s00454-015-9712-z54:3(663-668)Online publication date: 1-Oct-2015
https://dl.acm.org/doi/10.1007/s00454-015-9712-z
Colin de Verdière ÉGinot GGoaoc XDey TWhitesides S(2012)Multinerves and helly numbers of acyclic familiesProceedings of the twenty-eighth annual symposium on Computational geometry10.1145/2261250.2261282(209-218)Online publication date: 17-Jun-2012
https://dl.acm.org/doi/10.1145/2261250.2261282
Langberg MSchulman L(2009)Contraction and Expansion of Convex SetsDiscrete & Computational Geometry10.5555/3116261.311639642:4(594-614)Online publication date: 1-Dec-2009
https://dl.acm.org/doi/10.5555/3116261.3116396
Bárány IMatoušek J(2006)Berge's theorem, fractional Helly, and art galleriesDiscrete Mathematics10.1016/j.disc.2005.12.028306:19-20(2303-2313)Online publication date: 1-Oct-2006
https://dl.acm.org/doi/10.1016/j.disc.2005.12.028
Alon NKalai GMatouek JMeshulam R(2002)Transversal numbers for hypergraphs arising in geometryAdvances in Applied Mathematics10.1016/S0196-8858(02)00003-929:1(79-101)Online publication date: 1-Jul-2002
https://dl.acm.org/doi/10.1016/S0196-8858%2802%2900003-9
Alon N(2002)Covering a hypergraph of subgraphsDiscrete Mathematics10.1016/S0012-365X(02)00427-2257:2-3(249-254)Online publication date: 28-Nov-2002
https://dl.acm.org/doi/10.1016/S0012-365X%2802%2900427-2
Bern Eppstein (2002)Multivariate Regression DepthDiscrete & Computational Geometry10.1007/s00454-001-0092-128:1(1-17)Online publication date: 1-Jul-2002
https://dl.acm.org/doi/10.1007/s00454-001-0092-1
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