TY - CONF

T1 - On empty convex polygons in a planar point set

AU - Pinchasi, Rom

AU - Radoičić, Radoš

AU - Sharir, Micha

N1 - Funding Information:
Work on this paper has been supported by a grant from the US—Israeli Binational Science Foundation, by NSF Grants CCR-97-32101 and CCR-00-98246, by a grant from the Israel Science Fund (for a Center of Excellence in Geometric Computing), and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. Part of the work on this paper was performed when the authors have visited MSRI, Berkeley, in the Fall of 2003. A preliminary version of this paper has appeared in Proceedings of the 20th ACM Annual Symposium on Computational Geometry, 2004, pp. 391–400.

PY - 2004

Y1 - 2004

N2 - Let P be a set of n points in general position in the plane. Let X k(P) denote the number of empty convex k-gons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities Xk(P) and several related quantities. Most of these equalities and inequalities are new, except for a couple that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, algebraic topology and commutative algebra. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.

AB - Let P be a set of n points in general position in the plane. Let X k(P) denote the number of empty convex k-gons determined by P. We derive, using elementary proof techniques, several equalities and inequalities involving the quantities Xk(P) and several related quantities. Most of these equalities and inequalities are new, except for a couple that have been proved earlier using a considerably more complex machinery from matroid and polytope theory, algebraic topology and commutative algebra. Some of these relationships are also extended to higher dimensions. We present several implications of these relationships, and discuss their connection with several long-standing open problems, the most notorious of which is the existence of an empty convex hexagon in any point set with sufficiently many points.

KW - Continuous motion

KW - Empty hexagon

KW - Empty k-gon

UR - http://www.scopus.com/inward/record.url?scp=4544307565&partnerID=8YFLogxK

U2 - 10.1145/997817.997876

DO - 10.1145/997817.997876

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AN - SCOPUS:4544307565

SP - 391

EP - 400

T2 - Proceedings of the Twentieth Annual Symposium on Computational Geometry (SCG'04)

Y2 - 9 June 2004 through 11 June 2004

ER -