Unit distances in three dimensions

Haim Kaplan*, Jiří Matoušek, Zuzana Safernová, Micha Sharir

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We show that the number of unit distances determined by n points in R 3 is O(n 3/2), slightly improving the bound of Clarkson, Edelsbrunner, Guibas, Sharir and Welzl [5], established in 1990. The new proof uses the recently introduced polynomial partitioning technique of Guth and Katz [12]. While this paper was still in a draft stage, a similar proof of our main result was posted to the arXiv by Joshua Zahl [28].

Original languageEnglish
Pages (from-to)597-610
Number of pages14
JournalCombinatorics Probability and Computing
Issue number4
StatePublished - Jul 2012


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