distinct distances in three and higher dimensions
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 ?(n 77/141-? )=?(n 0.546 ) , for any ?>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.
distinct distances incidences point configurations