Volume 450 of the series Lecture Notes in Computer Science pp 419-428


A hyperplane Incidence problem with applications to counting distances

  • Herbert EdelsbrunnerAffiliated withDepartment of Computer Science, University of Illinois at Urbana-Champaign
  • , Micha SharirAffiliated withCourant Institute of Mathematical Sciences, New York UniversitySchool of Mathematical Sciences, Tel Aviv University

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This paper proves an O(m 2/3 n 2/3+m+n) upper bound on the number of incidences between m points and n hyperplanes in four dimensions, assuming all points lie on one side of each hyperplane and the points and hyperplanes satisfy certain natural general position conditions. This result has application to various three-dimensional combinatorial distance problems. For example, it implies the same upper bound for the number of bichromatic minimum distance pairs in a set of m blue and n red points in three-dimensional space. This improves the best previous bound for this problem.