@article{fd97fb46a8c542468f524831d761d9f3,

title = "On the number of directions determined by a three-dimensional points set",

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.",

author = "J{\'a}nos Pach and Rom Pinchasi and Micha Sharir",

note = "Funding Information: Work on this paper by J{\'a}nos Pach and Micha Sharir has been supported by NSF Grants CCR-97-32101 and CCR-00-98246, and by a joint grant from the US–Israel Binational Science Foundation. Work by J{\'a}nos Pach has also been supported by PSC-CUNY Research Award 63382-0032 and by Hungarian Science Foundation Grant OTKA T-043520. Work by Micha Sharir has also been supported by a grant from the Israeli Academy of Sciences for a Center of Excellence in Geometric Computing at Tel Aviv University, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University.",

year = "2004",

month = oct,

doi = "10.1016/j.jcta.2004.04.010",

language = "אנגלית",

volume = "108",

pages = "1--16",

journal = "Journal of Combinatorial Theory - Series A",

issn = "0097-3165",

publisher = "Academic Press Inc.",

number = "1",

}