A Non-linear Lower Bound for Planar Epsilon-nets

We show that the minimum possible size of an ε-net for point objects and line (or rectangle)-ranges in the plane is (slightly) bigger than linear in $\frac{1}{\epsilon}$. This settles a problem raised by Matoušek, Seidel and Welzl (Proc. 6th Annu. ACM Sympos. Comput. Geom., pp. 16–22, 1990).

[1]  János Pach,et al.  Combinatorial Geometry , 2012 .

[2]  Saurabh Ray,et al.  New existence proofs ε-nets , 2008, SCG '08.

[3]  Zoltán Füredi,et al.  Traces of finite sets: extremal problems and geometric applications , 1994 .

[4]  Jirí Matousek,et al.  New Constructions of Weak ε-Nets , 2004, Discret. Comput. Geom..

[5]  D. Polymath,et al.  A new proof of the density Hales-Jewett theorem , 2009, 0910.3926.

[6]  Michael T. Goodrich,et al.  Almost optimal set covers in finite VC-dimension , 1995, Discret. Comput. Geom..

[7]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[8]  Kenneth L. Clarkson,et al.  Improved Approximation Algorithms for Geometric Set Cover , 2005, Discret. Comput. Geom..

[9]  J. Pach,et al.  Wiley‐Interscience Series in Discrete Mathematics and Optimization , 2011 .

[10]  Gerhard J. Woeginger,et al.  Some new bounds for Epsilon-nets , 1990, SCG '90.

[11]  Leonidas J. Guibas,et al.  Improved bounds on weak ε-nets for convex sets , 1993, STOC.

[12]  János Komlós,et al.  Almost tight bounds forɛ-Nets , 1992, Discret. Comput. Geom..

[13]  J. Komlos,et al.  Almost tight bounds for $\epsilon$-nets , 1992 .

[14]  Noga Alon,et al.  The Probabilistic Method , 2015, Fundamentals of Ramsey Theory.

[15]  David G. Kirkpatrick,et al.  Improved Approximation for Guarding Simple Galleries from the Perimeter , 2010, Discret. Comput. Geom..

[16]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[17]  Jirí Matousek,et al.  Lower bounds for weak epsilon-nets and stair-convexity , 2008, SCG '09.

[18]  Kasturi R. Varadarajan.  Epsilon Nets and Union Complexity , 2009 .

[19]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[20]  Jiri Matousek,et al.  Lectures on discrete geometry , 2002, Graduate texts in mathematics.

[21]  Noga Alon,et al.  Point Selections and Weak ε-Nets for Convex Hulls , 1992, Combinatorics, Probability and Computing.

[22]  Jirí Matousek,et al.  Reporting Points in Halfspaces , 1992, Comput. Geom..

[23]  I. Bárány.  LECTURES ON DISCRETE GEOMETRY (Graduate Texts in Mathematics 212) , 2003 .

[24]  Noga Alon,et al.  Partitioning and geometric embedding of range spaces of finite Vapnik-Chervonenkis dimension , 1987, SCG '87.

[25]  BORIS ARONOV,et al.  Small-size ε-nets for axis-parallel rectangles and boxes , 2009, STOC '09.

[26]  J. Pach,et al.  Combinatorial geometry , 1995, Wiley-Interscience series in discrete mathematics and optimization.

[27]  D. Polymath,et al.  Density Hales-Jewett and Moser Numbers , 2010 .

[28]  Noga Alon,et al.  Transversal numbers for hypergraphs arising in geometry , 2002, Adv. Appl. Math..

[29]  J. van Leeuwen,et al.  Discrete and Computational Geometry , 2002, Lecture Notes in Computer Science.

[30]  A. Hales,et al.  Regularity and Positional Games , 1963 .

[31]  Micha Sharir,et al.  Small-Size $\eps$-Nets for Axis-Parallel Rectangles and Boxes , 2010, SIAM J. Comput..

[32]  János Pach,et al.  Indecomposable Coverings , 2005, Canadian Mathematical Bulletin.

[33]  Jirí Matousek,et al.  How to net a lot with little: small ε-nets for disks and halfspaces , 1990, SCG '90.

[34]  Noga Alon,et al.  The Probabilistic Method, Third Edition , 2008, Wiley-Interscience series in discrete mathematics and optimization.

[35]  H. Edelsbrunner,et al.  Improved bounds on weak ε-nets for convex sets , 1995, Discret. Comput. Geom..

[36]  H. Furstenberg,et al.  A density version of the Hales-Jewett theorem , 1991 .

[37]  Pankaj K. Agarwal.  Partitioning arrangements of lines I: An efficient deterministic algorithm , 1990, Discret. Comput. Geom..

[38]  Bernard Chazelle,et al.  A deterministic view of random sampling and its use in geometry , 1988, [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science.

[39]  David Haussler,et al.  ɛ-nets and simplex range queries , 1987, Discret. Comput. Geom..

[40]  Dror Rawitz,et al.  Hitting sets when the VC-dimension is small , 2005, Inf. Process. Lett..

[41]  Kenneth L. Clarkson,et al.  New applications of random sampling in computational geometry , 1987, Discret. Comput. Geom..

[42]  Y. Katznelson,et al.  A density version of the Hales-Jewett theorem for k=3 , 1989, Discret. Math..

[43]  N. Alon,et al.  The Probabilistic Method: Alon/Probabilistic , 2008 .