### 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 (US) |
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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 |

State | 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

Aronov, B., Pach, J., Sharir, M., & Tardos, G. (2003). Distinct distances in three and higher dimensions. In

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