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Maximal perimeter, diameter and area of equilateral unit-width convex polygons
2011
Journal of Global Optimization
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The paper also considers the maximization of the sum of distances between all pairs of vertices of equilateral unit-width convex polygons. ...
The paper answers the three distinct questions of maximizing the perimeter, diameter and area of equilateral unit-width convex polygons. ...
The values represent the upper and lower bounds of the width of an equilateral convex polygon with unit-perimeter, unit-diameter, unit-area or unit sum of distances. ...
doi:10.1007/s10898-011-9780-4
fatcat:4udnacaitrdc5d3r6ncxdnq7me
On Isosceles Triangles and Related Problems in a Convex Polygon
[article]
2010
arXiv
pre-print
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most n/k regular k-gons for any integer k> 4 and that this bound is optimal, and (iv) provide a short proof that the sum of all the distances between its vertices is at least (n-1)/2 and at most n/2 n/ ...
Given any convex n-gon, in this article, we: (i) prove that its vertices can form at most n^2/2 + Θ(n n) isosceles trianges with two sides of unit length and show that this bound is optimal in the first ...
In [3] , Altman proved that the number of distinct distances among all of the vertices of any convex n-gon is at least ⌊n/2⌋, a bound that is achieved by a regular polygon. ...
arXiv:1009.2218v2
fatcat:ltpbfqpvrzdi3a3ezbuv7ac4ue
Convexity Problems on Meshes with Multiple Broadcasting
1995
Journal of Parallel and Distributed Computing
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We show that the same time lower bound holds for the tasks of detecting whether a convex n-gon lies inside another as well as for computing the maximum distance between two convex n-gons. ...
Finally, we show that for two separable convex n-gons P and Q, the task of computing the minimum distance between P and Q can be performed in O(1) time on a mesh with multiple broadcasting of size n n. ...
Their constructive criticism and comments have resulted in a much improved presentation. We wish to thank Professor Sahni for his timely and professional handling of our submission. ...
doi:10.1006/jpdc.1995.1078
fatcat:addyhjklpnedzmssj2ht6frcii
Blind approximation of planar convex sets
1994
IEEE Transactions on Robotics and Automation
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A lower bound on the number of probings required by any strategy for achieving such an approximation is also derived. showing that the gap between the number of probings required by our strategy and the ...
A systematic probing strategy is suggested and an upper bound on the number of probings it requires for achieving an approximation with a pre-specified precision to the unknown object is derived. ...
All of them contributed a lot to improve this paper both in form and in substance. ...
doi:10.1109/70.313101
fatcat:opmbrvvw45cqpp36cloqhupwia
Two-Dimensional Range Diameter Queries
[chapter]
2012
Lecture Notes in Computer Science
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We strengthen the evidence by giving a lower bound for an important subproblem arising in solutions to the range diameter problem: computing the diameter of two convex polygons, that are separated by a ...
vertical line and are preprocessed independently, requires almost linear time in the number of vertices of the smaller polygon, no matter how much space is used. ...
We would like to thank Elad Verbin for introducing the set intersection problem, and Gerth Stølting Brodal, Jakob Truelsen, Konstantinos Tsakalidis, and Qin Zhang for informative discussions. ...
doi:10.1007/978-3-642-29344-3_19
fatcat:cib7ffpsrjejnfhn4pdge75scm
The maximum size of a convex polygon in a restricted set of points in the plane
1989
Discrete & Computational Geometry
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We show that there exist at least flk 1/4 of these points which are the vertices of a convex polygon, for some positive constant /3 =/3(a). ...
On the other hand, we show that for every fixed e>0, if k>k(e), then there is a set of k points in the plane for which the above ratio is at most 4~, which does not contain a convex polygon of more than ...
Lemma 3 . 1 . 31 Let P be a convex polygon on m vertices and let q denote the diameter of P, i.e., the maximum distance between a pair of vertices of P. ...
doi:10.1007/bf02187725
fatcat:df3d7i22ijg4dniamqfuyg7edy
Computational Geometry Column 34
[article]
1998
arXiv
pre-print
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Problems presented at the open-problem session of the 14th Annual ACM Symposium on Computational Geometry are listed. ...
Hajnal, A lower bound on the number of unit distances between the vertices of a convex polygon, J. Combin. Theory Ser. A 56 (1991), 312-316. [Fu] Z. ...
Füredi, The maximum number of unit distances in a convex n-gon, J. Combin. Theory Ser. A 55 (1990), 316-320. ...
arXiv:cs/9808008v1
fatcat:i26vqq277fe5fmp2f2fj5torwa
Multiplicities of interpoint distances in finite planar sets
1995
Discrete Applied Mathematics
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What is the maximum number of unit distances between the vertices of a convex n-gon in the plane? ...
We review known partial results for this and other open questions on multiple occurrences of the same interpoint distanc,~ in finite planar subsets. Some new results are proved for small n. ...
Acknowledgement We are indebted to a referee for the proof of Theorem 5 presented here. ...
doi:10.1016/0166-218x(94)00046-g
fatcat:tkdazraobrdutoihpiifhjokdq
Minimum Weight Convex Steiner Partitions
2009
Algorithmica
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This O(W log n) bound is the best possible due to Eppstein's lower bound on minimum weight Steiner triangulations [14] . ...
Without Steiner points, the corresponding bound is known to be Θ(log n), attained for n vertices of a pseudo-triangle. ...
We are indebted to Nadia Benbernou, Erik Demaine, Martin Demaine, Mashhood Ishaque, and Diane Souvaine for valuable conversations on these matters. ...
doi:10.1007/s00453-009-9329-9
fatcat:6rh6xmedofgwpb5r724lp23yyu
The Big Triangle Small Triangle Method for the Solution of Nonconvex Facility Location Problems
2004
Operations Research
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The resulting algorithm was tested on the obnoxious facility location and the attraction-repulsion Weber problems with excellent results. ...
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. ...
It is easier to calculate such a lower bound because its minimum is at one of the three vertices of the triangle, while a convex function may obtain its lower bound anywhere in the triangle. treated the ...
doi:10.1287/opre.1030.0077
fatcat:ff2bdw4wgfdj7elbcjmqhueam4
Extremal problems for convex polygons
2006
Journal of Global Optimization
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Consider a convex polygon V n with n sides, perimeter P n , diameter D n , area A n , sum of distances between vertices S n and width W n . ...
Minimizing or maximizing any of these quantities while fixing another defines ten pairs of extremal polygon problems (one of which usually has a trivial solution or no solution at all). ...
The sum S n of distances between vertices v 1 , v 2 , . . . , v n of a convex polygon with unit perimeter satisfies n − 1 2 < S n < 1 2 n 2 n 2 , the bounds being approached arbitrarily closely by the ...
doi:10.1007/s10898-006-9065-5
fatcat:3avhok322fdolhx5tsoc5qttoa
Interactive 3D distance field computation using linear factorization
2006
Proceedings of the 2006 symposium on Interactive 3D graphics and games - SI3D '06
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We also improve the performance by using culling techniques that reduce the number of distance function evaluations using bounds on Voronoi regions of the primitives. ...
Given a set of piecewise linear geometric primitives, our algorithm computes the distance field for each slice of a uniform spatial grid. ...
We thank the UNC GAMMA group for many useful discussions and support. We are also grateful to the reviewers for their feedback. ...
doi:10.1145/1111411.1111432
dblp:conf/si3d/SudGGM06
fatcat:rzq4umfaw5a6blundntmxripuy
The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees
1983
Journal of the ACM
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When the input is a convex n-vertex polygon rather than an arbitrary set of points the Ω(n log n) lower bound does not hold. ...
A more difficult problem for the case of convex polygons calls for finding the closest pair of vertices p i , q j between two convex polygons P = (p 1 , p 2 ,..., p n ) and Q = (q 1 , q 2 ,..., q n ), ...
Furthermore, when the sets form a convex polygon this complexity can be reduced to O(n). ...
doi:10.1145/2402.322386
fatcat:knncv4rvw5ea7f4hre5lr2x5gq
A compact piecewise-linear voronoi diagram for convex sites in the plane
1996
Discrete & Computational Geometry
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If these sets are polygons with n total vertices, we compute this diagram optimally in O( k log n) deterministic time for the Euclidean metric and in O(k logn logm) deterministic time for the convex distance ...
I n the plane, the post-ofice problem, which asks for the closest site to a query site, and retraction motion planning, which asks for a one-dimensional retract of the free space of a robot, are both classtcally ...
Acknowledgments We thank Stephan Meiser for discussions on the randomized incremental construction of our compact diagrams. ...
doi:10.1007/bf02716580
fatcat:nepxlerf5nhxbdqywjvlhgiege
Linear approximation of simple objects
1994
Computational geometry
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Let S be a family of m convex polygons in the plane with a total number of n vertices and let each polygon have a positive weight associated with it. ...
For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O(nm logm) time and O(n) space. ...
Acknowledgement We like to thank an anonymous referee and Katrin Dobrindt who pointed out how to reduce the number of candidates for the MMA problem in E3. ...
doi:10.1016/0925-7721(94)90015-9
fatcat:ku2mdvrmlbg6thxdhrelyur6jm
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