### Abstract

According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size. Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1/ε, were found by Alon (2010). In his examples, every ε-net is of size, where g is an extremely slowly growing function, related to the inverse Ackermann function. We show that there exist geometrically defined range spaces, already of VC-dimension, in which the size of the smallest ε-nets is. We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is. By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

Original language | English |
---|---|

Pages (from-to) | 645-658 |

Number of pages | 14 |

Journal | Journal of the American Mathematical Society |

Volume | 26 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2013 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

**Tight lower bounds for the size of epsilon-nets.** / Pach, János; Tardos, G.

Research output: Contribution to journal › Article

*Journal of the American Mathematical Society*, vol. 26, no. 3, pp. 645-658. https://doi.org/10.1090/S0894-0347-2012-00759-0

}

TY - JOUR

T1 - Tight lower bounds for the size of epsilon-nets

AU - Pach, János

AU - Tardos, G.

PY - 2013

Y1 - 2013

N2 - According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size. Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1/ε, were found by Alon (2010). In his examples, every ε-net is of size, where g is an extremely slowly growing function, related to the inverse Ackermann function. We show that there exist geometrically defined range spaces, already of VC-dimension, in which the size of the smallest ε-nets is. We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is. By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

AB - According to a well known theorem of Haussler and Welzl (1987), any range space of bounded VC-dimension admits an ε-net of size. Using probabilistic techniques, Pach and Woeginger (1990) showed that there exist range spaces of VC-dimension 2, for which the above bound is sharp. The only known range spaces of small VC-dimension, in which the ranges are geometric objects in some Euclidean space and the size of the smallest ε-nets is superlinear in 1/ε, were found by Alon (2010). In his examples, every ε-net is of size, where g is an extremely slowly growing function, related to the inverse Ackermann function. We show that there exist geometrically defined range spaces, already of VC-dimension, in which the size of the smallest ε-nets is. We also construct range spaces induced by axis-parallel rectangles in the plane, in which the size of the smallest ε-nets is. By a theorem of Aronov, Ezra, and Sharir (2010), this bound is tight.

UR - http://www.scopus.com/inward/record.url?scp=84878709119&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84878709119&partnerID=8YFLogxK

U2 - 10.1090/S0894-0347-2012-00759-0

DO - 10.1090/S0894-0347-2012-00759-0

M3 - Article

VL - 26

SP - 645

EP - 658

JO - Journal of the American Mathematical Society

JF - Journal of the American Mathematical Society

SN - 0894-0347

IS - 3

ER -