Distinct distances in three and higher dimensions

Boris Aronov; János Pach; Micha Sharir; Gábor Tardos

doi:10.1145/780542.780621

Distinct distances in three and higher dimensions

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

^*Corresponding author for this work

School of Computer Science

Research output: Contribution to journal › Conference article › peer-review

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
Pages (from-to)	541-546
Number of pages	6
Journal	Conference Proceedings of the Annual ACM Symposium on Theory of Computing
DOIs	https://doi.org/10.1145/780542.780621
State	Published - 2003
Event	35th Annual ACM Symposium on Theory of Computing - San Diego, CA, United States Duration: 9 Jun 2003 → 11 Jun 2003

Keywords

Distinct distances
Incidences
Point configurations

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10.1145/780542.780621

Cite this

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title = "Distinct distances in three and higher dimensions",

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TY - JOUR

T1 - Distinct distances in three and higher dimensions

AU - Aronov, Boris

AU - Pach, János

AU - Sharir, Micha

AU - Tardos, Gábor

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

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

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

Y2 - 9 June 2003 through 11 June 2003

ER -

Distinct distances in three and higher dimensions

Abstract

Keywords

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