### Abstract

Given a set P of n points in R^{d} and ε> 0, we consider the problem of constructing weak ε-nets for P. We show the following: pick a random sample Q of size O(1/ε log(1/ε)) from P. Then, with constant probability, a weak ε-net of P can be constructed from only the points of Q . This shows that weak ε-nets in Rd can be computed from a subset of P of size O(1/ε log(1/ε)) with only the constant of proportionality depending on the dimension, unlike all previous work where the size of the subset had the dimension in the exponent of 1/ε. However, our final weak ε-nets still have a large size (with the dimension appearing in the exponent of 1/ε).

Original language | English (US) |
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Pages (from-to) | 565-571 |

Number of pages | 7 |

Journal | Computational Geometry: Theory and Applications |

Volume | 43 |

Issue number | 6-7 |

DOIs | |

State | Published - Aug 2010 |

### Keywords

- Combinatorial geometry
- Hitting convex sets
- Weak ε-nets

### ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics

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## Cite this

Mustafa, N. H., & Ray, S. (2010). Reprint of: Weak ε-nets have basis of size.

*Computational Geometry: Theory and Applications*,*43*(6-7), 565-571. https://doi.org/10.1016/j.comgeo.2007.02.007