Distinct distances in three and higher dimensions

Authors:
Boris Aronov;János Pach;Micha Sharir;Gábor Tardos
Affiliations:
Polytechnic University, Brooklyn, NY;New York University, New York, NY;Tel Aviv University, Tel Aviv Israel and New York University, New York, NY;Rényi Institute, Hungarian Academy of Sciences, Pf. 127, H-1354 Budapest, Hungary
Venue:
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Year:
2003

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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 ε0. 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.