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Distinct Distances in Three and Higher Dimensions

Published online by Cambridge University Press:  28 April 2004

BORIS ARONOV
Affiliation:
Department of Computer and Information Science, Polytechnic University, Brooklyn, NY 11201-3840, USA (e-mail: aronov@cis.poly.edu)
JÁNOS PACH
Affiliation:
City College, CUNY and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA (e-mail: pach@cims.nyu.edu)
MICHA SHARIR
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel, and Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA (e-mail: michas@post.tau.ac.il)
GÁBOR TARDOS
Affiliation:
Rényi Institute, Hungarian Academy of Sciences, Pf. 127, H-1354 Budapest, Hungary (e-mail: tardos@renyi.hu)

Abstract

Improving an old result of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl, we show that the number of distinct distances determined by a set $P$ of $n$ points in three-dimensional space is $\Omega(n^{77/141-\varepsilon})=\Omega(n^{0.546})$, for any $\varepsilon>0$. Moreover, there always exists a point $p\in P$ from which there are at least so 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.

Type
Paper
Copyright
2004 Cambridge University Press

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