 
Summary: Distinct Distances in Three and Higher Dimensions #
Boris Aronov + J’anos Pach # Micha Sharir § G’abor Tardos ¶
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 threedimensional space is # n 77/141# ) =
# n 0.546 ), for any # > 0. Moreover, there always exists a point p # 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 threedimensional sphere. As a consequence, we obtain
analogous results in higher dimensions.
1 Introduction
``My most striking contribution to geometry is, no doubt, my problem on the number of
distinct distances''  wrote Erdťos on his 80th birthday [8]. What is the minimum number of
distinct interpoint distances determined by n points in R d ? More precisely, Erdťos [7] asked
the following question in 1946. Given a point set P , let g(P ) denote the number of distinct
distances between the elements of P . Let g d (n) = min P g(P ), where the minimum is taken
over all sets P of n distinct points in dspace. We want to describe the asymptotic behavior
of the function g d (n). More than 50 years later, in spite of considerable e#orts, we are still far
from knowing the correct order of magnitude of g d (n) even in the plane (d = 2). This problem
is more than just a ``gem'' in recreational mathematics. It was an important motivating
# Work on this paper has been supported by a grant from the U.S.Israeli Binational Science Foundation.
