Distinct distances in three and higher dimensions

Boris Aronov*, János Pach, Micha Sharir, Gábor Tardos

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review


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.

Original languageEnglish
Pages (from-to)541-546
Number of pages6
JournalConference Proceedings of the Annual ACM Symposium on Theory of Computing
StatePublished - 2003
Event35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States
Duration: 9 Jun 200311 Jun 2003


  • Distinct distances
  • Incidences
  • Point configurations

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