On empty convex polygons in a planar point set
نویسندگان
چکیده
منابع مشابه
Empty convex polygons in planar point sets
The Erdős–Szekeres theorem inspired a lot of research. A frequent topic in this area is the study of the existence of so-called empty convex polygons in finite planar point sets. Let P be a finite set of points in general position in the plane. A convex k-gon G is called a k-hole (or empty convex k-gon) of P , if all vertices of G lie in P and no point of P lies inside G. Frequently we will mea...
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Erdős asked whether every sufficiently large set of points in general position in the plane contains six points that form a convex hexagon without any points from the set in its interior. Such a configuration is called an empty convex hexagon. In this paper, we answer the question in the affirmative. We show that every set that contains the vertex set of a convex 9-gon also contains an empty co...
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Given a set S of n points in the plane, we compute in time O(n) the total number of convex polygons whose vertices are a subset of S. We give an O(m n) algorithm for computing the number of convex k-gons with vertices in S, for all values k = 3; : : : ;m; previously known bounds were exponential (O(ndk=2e)). We also compute the number of empty convex polygons (resp., k-gons, k m) with vertices ...
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A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P . We construct a set of n points in general position in the plane with only ≈ 1.62n empty triangles, ≈ 1.94n empty quadrilaterals, ≈ 1.02n empty pentagons, and ≈ 0.2n empty hexagons.
متن کاملDisjoint empty convex pentagons in planar point sets
Harborth [Elemente der Mathematik, Vol. 33 (5), 116–118, 1978] proved that every set of 10 points in the plane, no three on a line, contains an empty convex pentagon. From this it follows that the number of disjoint empty convex pentagons in any set of n points in the plane is least ⌊ n 10 ⌋. In this paper we prove that every set of 19 points in the plane, no three on a line, contains two disjo...
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ژورنال
عنوان ژورنال: Journal of Combinatorial Theory, Series A
سال: 2006
ISSN: 0097-3165
DOI: 10.1016/j.jcta.2005.03.007