### Abstract

Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in three-dimensional space is Ω(n^{77/141-ε}) = Ω(n^{0.546}), for any ε > 0. Moreover, there always exists a point p ∈ P from which there are at least these many distinct distances to the remaining elements of P. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions.

Original language | English |
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Title of host publication | Conference Proceedings of the Annual ACM Symposium on Theory of Computing |

Pages | 541-546 |

Number of pages | 6 |

Publication status | Published - 2003 |

Event | 35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: Jun 9 2003 → Jun 11 2003 |

### Other

Other | 35th Annual ACM Symposium on Theory of Computing |
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Country | United States |

City | San Diego, CA |

Period | 6/9/03 → 6/11/03 |

### Keywords

- Distinct distances
- Incidences
- Point configurations

### ASJC Scopus subject areas

- Software

### Cite this

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing*(pp. 541-546)

**Distinct distances in three and higher dimensions.** / Aronov, Boris; Pach, János; Sharir, Micha; Tardos, G.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Conference Proceedings of the Annual ACM Symposium on Theory of Computing.*pp. 541-546, 35th Annual ACM Symposium on Theory of Computing, San Diego, CA, United States, 6/9/03.

}

TY - GEN

T1 - Distinct distances in three and higher dimensions

AU - Aronov, Boris

AU - Pach, János

AU - Sharir, Micha

AU - Tardos, G.

PY - 2003

Y1 - 2003

N2 - Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in three-dimensional space is Ω(n77/141-ε) = Ω(n0.546), for any ε > 0. Moreover, there always exists a point p ∈ P from which there are at least these many distinct distances to the remaining elements of P. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions.

AB - Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in three-dimensional space is Ω(n77/141-ε) = Ω(n0.546), for any ε > 0. Moreover, there always exists a point p ∈ P from which there are at least these many distinct distances to the remaining elements of P. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous results in higher dimensions.

KW - Distinct distances

KW - Incidences

KW - Point configurations

UR - http://www.scopus.com/inward/record.url?scp=0038107574&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0038107574&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:0038107574

SP - 541

EP - 546

BT - Conference Proceedings of the Annual ACM Symposium on Theory of Computing

ER -