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Convex position

About: Convex position is a(n) research topic. Over the lifetime, 354 publication(s) have been published within this topic receiving 5265 citation(s). more


Open accessBook ChapterDOI: 10.1007/978-0-8176-4842-8_3
Abstract: Our present problem has been suggested by Miss Esther Klein in connection with the following proposition. more

Topics: Convex geometry (67%), Connection (mathematics) (67%), Discrete geometry (63%) more

1,467 Citations

Proceedings ArticleDOI: 10.1109/SFCS.2000.892132
Ileana Streinu1Institutions (1)
12 Nov 2000-
Abstract: We propose a combinatorial approach to plan noncolliding motions for a polygonal bar-and-joint framework. Our approach yields very efficient deterministic algorithms for a category of robot arm motion planning problems with many degrees of freedom, where the known general roadmap techniques would give exponential complexity. It is based on a novel class of one-degree-of-freedom mechanisms induced by pseudo triangulations of planar point sets, for which we provide several equivalent characterization and exhibit rich combinatorial and rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter's rule problem: convexify a simple bar-and-joint planar polygonal linkage using only non self-intersecting planar motions. A step in the convexification motion consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At that point, a local alteration restores the pseudo triangulation. The motion continues for O(n/sup 2/) steps until all the points are in convex position. more

Topics: Motion planning (55%), Robotic arm (52%), Configuration space (51%) more

182 Citations

Open accessJournal ArticleDOI: 10.1016/S0925-7721(00)00010-9
Alfredo García1, Marc Noy2, Javier Tejel1Institutions (2)
Abstract: We improve previous lower bounds on the number of simple polygonizations, and other kinds of crossing-free subgraphs, of a set of N points in the plane by analyzing a suitable configuration. We also prove that the number of crossing-free perfect matchings and spanning trees is minimum when the points are in convex position. more

135 Citations

Book ChapterDOI: 10.1007/978-3-642-58043-7_6
01 Jan 1993-
Abstract: The allowable sequence associated to a configuration of points was first developed by the authors in order to investigate what combinatorial structure lay behind the Erdős-Szekeres conjecture (that any 2 n-2 + 1 points in general position in the plane contain among them n points which are in convex position) Though allowable sequences did not lead to any progress on this ancient problem, there did emerge an object that had considerable intrinsic interest, that turned out to be related to some other well-studied structures such as pseudoline arrangements and oriented matroids, and that had as well a combinatorial simplicity and suggestiveness which turned out to be effective in the solution of several other classical problems These connections and applications are discussed in Sections 2, 3, and 4 of this paper more

Topics: Convex position (55%), General position (53%), Computational geometry (52%) more

128 Citations

Open accessJournal ArticleDOI: 10.1090/S0273-0979-00-00877-6
W. Morris1, V. Soltan1Institutions (1)
Abstract: In 1935 Erdős and Szekeres proved that for any integer n ≥ 3 there exists a smallest positive integer N(n) such that any set of at least N(n) points in general position in the plane contains n points that are the vertices of a convex n-gon. They also posed the problem to determine the value of N(n) and conjectured that N(n) = 2n−2 + 1 for all n ≥ 3. Despite the efforts of many mathematicians, the Erdős-Szekeres problem is still far from being solved. This paper surveys the known results and questions related to the Erdős-Szekeres problem in the plane and higher dimensions, as well as its generalizations for the cases of families of convex bodies and the abstract convexity setting. more

Topics: Convex position (64%), Convex set (58%), Convex combination (56%) more

123 Citations

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Topic's top 5 most impactful authors

Pavel Valtr

22 papers, 296 citations

Oswin Aichholzer

18 papers, 190 citations

Michiel Smid

17 papers, 148 citations

Thomas Hackl

14 papers, 146 citations

Adrian Dumitrescu

14 papers, 78 citations

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