Distinct distances in three and higher dimensions

  title={Distinct distances in three and higher dimensions},
  author={Boris Aronov and J{\'a}nos Pach and Micha Sharir and G{\'a}bor Tardos},
Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set <i>P</i> of <i>n</i> points in three-dimensional space is <i>Ω(n<sup>77/141-ε</sup>)=Ω(n<sup>0.546</sup>)</i>, for any <i>ε>0</i>. Moreover, there always exists a point <i>p ∈ P</i> from which there are at least these many distinct distances to the remaining elements of <i>P</i>. The same result holds for points on the three-dimensional sphere. As a consequence, we obtain analogous… CONTINUE READING
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