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On empty convex polygons in a planar point set

Rom Pinchasi, Radoš Radoičić, Micha Sharir
2006 Journal of combinatorial theory. Series A  
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.  ...  any point set with sufficiently many points.  ...  works on these problems.  ... 
doi:10.1016/j.jcta.2005.03.007 fatcat:dbb5z6yykrberolwhswbalxqze

On empty convex polygons in a planar point set

Rom Pinchasi, Rados Radoicic, Micha Sharir
2004 Proceedings of the twentieth annual symposium on Computational geometry - SCG '04  
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 .  ...  any point set with sufficiently many points.  ...  Acknowledgments The authors wish to thank Imre Bárány, Pavel Valtr and Boris Aronov for useful discussions and comments on the paper, and Paul Edelman for providing helpful information about the  ... 
doi:10.1145/997817.997876 dblp:conf/compgeom/PinchasiRS04 fatcat:yfmvbvrqzfbcvfvz2djkc7rwve

On Finding Large Polygonal Voids Using Delaunay Triangulation: The Case of Planar Point Sets [chapter]

Carlos Hervías, Nancy Hitschfeld-Kahler, Luis E. Campusano, Giselle Font
2014 Proceedings of the 22nd International Meshing Roundtable  
A polygon is considered a void if its area is larger than a threshold value. The algorithm is validated in 2D points with artificially generated circular and non-convex polygon voids.  ...  This paper proposes a new method to find voids that starting from local longest-edges in a Delaunay triangulation builds the largest possible empty or almost empty polygons around them.  ...  CH, NH and LEC received partial support from a CONICYT Anillo project (ACT 1122). NH was also partially supported by Fondecyt Project N 1120495.  ... 
doi:10.1007/978-3-319-02335-9_16 dblp:conf/imr/HerviasHCF13 fatcat:pfmqi4igrne3ncl7k4gkprnd6m

On the Reflexivity of Point Sets [chapter]

Esther M. Arkin, Sándor P. Fekete, Ferran Hurtado, Joseph S. B. Mitchell, Marc Noy, Vera Sacristán, Saurabh Sethia
2001 Lecture Notes in Computer Science  
We introduce a new measure for planar point sets Ë.  ...  Intuitively, it describes the combinatorial distance from a convex set: The reflexivity ´Ëµ of Ë is given by the smallest number of reflex vertices in a simple polygonization of Ë.  ...  The study of convex chains in finite planar point sets is the topic of classical papers by Erdős and Szekeres [7, 8] , who showed that any point set of size Ò has a convex subset of size Ø ª´ÐÓ Òµ.  ... 
doi:10.1007/3-540-44634-6_18 fatcat:jwrkuhg6a5e5rlvd6fkq42s4ci

On an empty triangle with the maximum area in planar point sets

Kiyoshi Hosono
2011 Discrete Mathematics  
Introduction We deal with only finite point sets P in the plane in general position. A point set is convex or in convex position if it determines a convex polygon.  ...  Let P be an n planar point set in general position. For a subset Q of P , denote the area of the convex hull of Q by A(Q).  ... 
doi:10.1016/j.disc.2011.05.006 fatcat:4kdyyo32pvh55pzibdntm6b3fu

Minimum Convex Partitions and Maximum Empty Polytopes [article]

Adrian Dumitrescu, Sariel Har-Peled, Csaba D. Tóth
2014 arXiv   pre-print
Let S be a set of n points in R^d. A Steiner convex partition is a tiling of conv(S) with empty convex bodies.  ...  We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.  ...  Many thanks also go to Sergio Cabello and Maria Saumell for pointing us to the recent results of Bautista-Santiago et al.  ... 
arXiv:1112.1124v4 fatcat:5iekym6skrchzp7hwcs3kkszau

Minimum convex partition of a constrained point set

Thomas Fevens, Henk Meijer, David Rappaport
2001 Discrete Applied Mathematics  
In this paper, we will present a polynomial time algorithm to ÿnd a minimum convex partition with respect to a point set S where S is constrained to lie on the boundaries of a ÿxed number of nested convex  ...  A convex partition with respect to a point set S is a planar subdivision whose vertices are the points of S, where the boundary of the unbounded outer face is the boundary of the convex hull of S, and  ...  A convex partition of P is a planar subdivision of P whose vertices are S ∩ P and where every face is a convex polygon. Deÿnition 2.8. Let P be a simple polygon and S be a point set in the plane.  ... 
doi:10.1016/s0166-218x(00)00237-7 fatcat:o2luujqinbbh3odzwn3tnaterq

Discretization of Planar Geometric Cover Problems [article]

Dae-Sung Jang, Han-Lim Choi
2014 arXiv   pre-print
We consider discretization of the 'geometric cover problem' in the plane: Given a set P of n points in the plane and a compact planar object T_0, find a minimum cardinality collection of planar translates  ...  We propose a reduced finite solution space that consists of distinct canonical translates and present polynomial algorithms to find the reduce solution space for disks, convex/non-convex polygons (including  ...  If the prototype is a convex polygon of m vertices, the inverses of the translates whose reference points are the points in a given set P are also convex polygons of m vertices.  ... 
arXiv:1411.6810v1 fatcat:cj2ega664ves3dyqwm7ge7f4na

Minimum Convex Partitions and Maximum Empty Polytopes [chapter]

Adrian Dumitrescu, Sariel Har-Peled, Csaba D. Tóth
2012 Lecture Notes in Computer Science  
Let S be a set of n points in R d . A Steiner convex partition is a tiling of conv(S) with empty convex bodies.  ...  We also give the first constant-factor approximation algorithm for computing a minimum Steiner convex partition of a planar point set in general position.  ...  Many thanks also go to Sergio Cabello and Maria Saumell for pointing us to the recent results of Bautista-Santiago et al.  ... 
doi:10.1007/978-3-642-31155-0_19 fatcat:uent4rh6szb43jvoe6p5lgae2e

Page 5125 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews  
Introduction: “We consider the following problem, called ‘batched planar point location’: Given a planar subdivision R induced by a plane graph G = (V, E), with |V| = N, and a set S of M points in the  ...  A sorting method is proposed that is based on the decomposition of a plane algebraic curve into convex segments and on point location in this decomposition.  ... 

On pseudo-convex decompositions, partitions, and coverings

Oswin Aichholzer, Clemens Huemer, Sarah Renkl, Bettina Speckmann, Csaba D. Tóth
2005 European Workshop on Computational Geometry  
We introduce pseudo-convex decompositions, partitions, and coverings for planar point sets. They are natural extensions of their convex counterparts and use both convex polygons and pseudo-triangles.  ...  We discuss some of their basic combinatorial properties and establish upper and lower bounds on their complexity.  ...  The first two authors want to thank Ferran Hurtado and Hannes Krasser for valuable discussions on the presented subject.  ... 
dblp:conf/ewcg/AichholzerHRST05 fatcat:s3i7qvbojzgv5ftv4wvncglxgy

On the Reflexivity of Point Sets [chapter]

Esther M. Arkin, Joseph S. B. Mitchell, Sándor P. Fekete, Ferran Hurtado, Marc Noy, Vera Sacristán, Saurabh Sethia
2003 Algorithms and Combinatorics  
in a simple polygonalization of S.  ...  We prove combinatorial bounds on the reflexivity of point sets and study some closely related quantities, including the convex cover number κc(S) of a planar point set, which is the smallest number of  ...  Acknowledgments We thank Adrian Dumitrescu for valuable input on this work, including a software tool for calculating reflexivity of point sets.  ... 
doi:10.1007/978-3-642-55566-4_6 fatcat:t2aghlo45jhgpgkycvqb6qxpti

EMPTY CONVEX 5-GONS IN PLANAR POINT SETS

Seong-Yoon Ann, En-Sil Kang
2009 Honam Mathematical Journal  
Erdös posed the problem of determining the minimum number g(n) such that any set of g(n) points in general position in the plane contains an empty convex n-gon.  ...  In 1978, Harborth proved that g(5) = 10. We reprove the result in a geometric approach.  ...  In this paper, we reprove a known result related to the Erdös and Szekeres problem which is a classic problem counting empty convex k-gons in planar points sets.  ... 
doi:10.5831/hmj.2009.31.3.315 fatcat:pgnvpinix5dv3cqjnjumj3junq

On convex closed planar curves as equidistant sets [article]

Csaba Vincze
2018 arXiv   pre-print
The equidistant set of two nonempty subsets K and L in the Euclidean plane is a set all of whose points have the same distance from K and L.  ...  In this sense the equidistancy is a generalization of the convexity.  ...  In general there are lots of different ways to give a convex polygon as an equidistant set.  ... 
arXiv:1705.07119v2 fatcat:mu72inf2k5adxdnophzr4fy6x4

Page 10054 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
Summary: “Given a planar point set in general position, S, we seek a partition of the points into convex cells, such that the union of the cells forms a simple polygon, P, and every point from S is on  ...  The author investigates a problem closely related to the Erdos- Szekeres theorem on finding points in convex position in planar point sets.  ... 
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