### Abstract

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 on the first selection lemma in ℝ^{3}. In particular, we prove that c_{3}≥0. 00227, improving the previous best result of c_{3}≥0. 00162 by Wagner (On k-sets and applications. Ph. D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c_{3}≤1/4^{4}≈0. 00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69-77, 1984) (where the two-dimensional case was settled).

Original language | English (US) |
---|---|

Pages (from-to) | 637-644 |

Number of pages | 8 |

Journal | Discrete and Computational Geometry |

Volume | 44 |

Issue number | 3 |

DOIs | |

State | Published - May 28 2010 |

### Fingerprint

### Keywords

- Centerpoint
- Selection lemma
- Simplex

### ASJC Scopus subject areas

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

### Cite this

^{3}

*Discrete and Computational Geometry*,

*44*(3), 637-644. https://doi.org/10.1007/s00454-010-9263-2

**Hitting Simplices with Points in ℝ ^{3} .** / Basit, Abdul; Mustafa, Nabil H.; Ray, Saurabh; Raza, Sarfraz.

Research output: Contribution to journal › Article

^{3}',

*Discrete and Computational Geometry*, vol. 44, no. 3, pp. 637-644. https://doi.org/10.1007/s00454-010-9263-2

^{3}Discrete and Computational Geometry. 2010 May 28;44(3):637-644. https://doi.org/10.1007/s00454-010-9263-2

}

TY - JOUR

T1 - Hitting Simplices with Points in ℝ3

AU - Basit, Abdul

AU - Mustafa, Nabil H.

AU - Ray, Saurabh

AU - Raza, Sarfraz

PY - 2010/5/28

Y1 - 2010/5/28

N2 - The so-called first selection lemma states the following: given any set P of n points in ℝd, there exists a point in ℝd contained in at least cdnd+1-O(nd) simplices spanned by P, where the constant cd depends on d. We present improved bounds on the first selection lemma in ℝ3. In particular, we prove that c3≥0. 00227, improving the previous best result of c3≥0. 00162 by Wagner (On k-sets and applications. Ph. D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c3≤1/44≈0. 00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69-77, 1984) (where the two-dimensional case was settled).

AB - The so-called first selection lemma states the following: given any set P of n points in ℝd, there exists a point in ℝd contained in at least cdnd+1-O(nd) simplices spanned by P, where the constant cd depends on d. We present improved bounds on the first selection lemma in ℝ3. In particular, we prove that c3≥0. 00227, improving the previous best result of c3≥0. 00162 by Wagner (On k-sets and applications. Ph. D. thesis, ETH Zurich, 2003). This makes progress, for the three-dimensional case, on the open problems of Bukh et al. (Stabbing simplices by points and flats. Discrete Comput. Geom., 2010) (where it is proven that c3≤1/44≈0. 00390) and Boros and Füredi (The number of triangles covering the center of an n-set. Geom. Dedic. 17(1):69-77, 1984) (where the two-dimensional case was settled).

KW - Centerpoint

KW - Selection lemma

KW - Simplex

UR - http://www.scopus.com/inward/record.url?scp=77955771874&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77955771874&partnerID=8YFLogxK

U2 - 10.1007/s00454-010-9263-2

DO - 10.1007/s00454-010-9263-2

M3 - Article

AN - SCOPUS:77955771874

VL - 44

SP - 637

EP - 644

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 3

ER -