Distinct distances in three and higher dimensions

Boris Aronov; János Pach; Micha Sharir; Gábor Tardos

Distinct distances in three and higher dimensions

Boris Aronov, János Pach, Micha Sharir, Gábor Tardos

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

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)
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
Country/Territory	United States
City	San Diego, CA
Period	6/9/03 → 6/11/03

Keywords

Distinct distances
Incidences
Point configurations

ASJC Scopus subject areas

Software

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

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(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.",

keywords = "Distinct distances, Incidences, Point configurations",

author = "Boris Aronov and J{\'a}nos Pach and Micha Sharir and G{\'a}bor Tardos",

year = "2003",

language = "English (US)",

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note = "35th Annual ACM Symposium on Theory of Computing ; Conference date: 09-06-2003 Through 11-06-2003",

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TY - GEN

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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M3 - Conference contribution

AN - SCOPUS:0038107574

SP - 541

EP - 546

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

T2 - 35th Annual ACM Symposium on Theory of Computing

Y2 - 9 June 2003 through 11 June 2003

ER -

Distinct distances in three and higher dimensions

Abstract

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Keywords

ASJC Scopus subject areas

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