On the number of directions determined by a three-dimensional points set

János Pach, Rom Pinchasi, Micha Sharir

Research output: Contribution to journalArticlepeer-review

Abstract

Let P be a set of n points in ℝ3, not all of which are in a plane and no three on a line. We partially answer a question of Scott (Amer. Math. Monthly 77 (1970) 502) by showing that the connecting lines of P assume at least 2n - 3 different directions if n is even and at least 2n - 2 if n is odd. These bounds are sharp. The proof is based on a far-reaching generalization of Ungar's theorem concerning the analogous problem in the plane.

Original languageEnglish
Pages (from-to)1-16
Number of pages16
JournalJournal of Combinatorial Theory - Series A
Volume108
Issue number1
DOIs
StatePublished - Oct 2004

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