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## Filters

##
###
Maximal perimeter, diameter and area of equilateral unit-width convex polygons

2011
*
Journal of Global Optimization
*

*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*. ...

##
###
On Isosceles Triangles and Related Problems in a Convex Polygon
[article]

2010
*
arXiv
*
pre-print

most n/k regular k-gons for any integer k> 4 and that this

arXiv:1009.2218v2
fatcat:ltpbfqpvrzdi3a3ezbuv7ac4ue
*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*. ...##
###
Convexity Problems on Meshes with Multiple Broadcasting

1995
*
Journal of Parallel and Distributed Computing
*

We show that

doi:10.1006/jpdc.1995.1078
fatcat:addyhjklpnedzmssj2ht6frcii
*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. ...##
###
Blind approximation of planar convex sets

1994
*
IEEE Transactions on Robotics and Automation
*

*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. ...

##
###
Two-Dimensional Range Diameter Queries
[chapter]

2012
*
Lecture Notes in Computer Science
*

We strengthen

doi:10.1007/978-3-642-29344-3_19
fatcat:cib7ffpsrjejnfhn4pdge75scm
*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. ...##
###
The maximum size of a convex polygon in a restricted set of points in the plane

1989
*
Discrete & Computational Geometry
*

We show that there exist at least flk 1/4

doi:10.1007/bf02187725
fatcat:df3d7i22ijg4dniamqfuyg7edy
*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. ...##
###
Computational Geometry Column 34
[article]

1998
*
arXiv
*
pre-print

Problems presented at

arXiv:cs/9808008v1
fatcat:i26vqq277fe5fmp2f2fj5torwa
*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. ...##
###
Multiplicities of interpoint distances in finite planar sets

1995
*
Discrete Applied Mathematics
*

What is

doi:10.1016/0166-218x(94)00046-g
fatcat:tkdazraobrdutoihpiifhjokdq
*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. ...##
###
Minimum Weight Convex Steiner Partitions

2009
*
Algorithmica
*

This O(W log n)

doi:10.1007/s00453-009-9329-9
fatcat:6rh6xmedofgwpb5r724lp23yyu
*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. ...##
###
The Big Triangle Small Triangle Method for the Solution of Nonconvex Facility Location Problems

2004
*
Operations Research
*

*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*...

##
###
Extremal problems for convex polygons

2006
*
Journal of Global Optimization
*

Consider

doi:10.1007/s10898-006-9065-5
fatcat:3avhok322fdolhx5tsoc5qttoa
*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*...##
###
Interactive 3D distance field computation using linear factorization

2006
*
Proceedings of the 2006 symposium on Interactive 3D graphics and games - SI3D '06
*

We also improve

doi:10.1145/1111411.1111432
dblp:conf/si3d/SudGGM06
fatcat:rzq4umfaw5a6blundntmxripuy
*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. ...##
###
The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees

1983
*
Journal of the ACM
*

When

doi:10.1145/2402.322386
fatcat:knncv4rvw5ea7f4hre5lr2x5gq
*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). ...##
###
A compact piecewise-linear voronoi diagram for convex sites in the plane

1996
*
Discrete & Computational Geometry
*

If these sets are

doi:10.1007/bf02716580
fatcat:nepxlerf5nhxbdqywjvlhgiege
*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. ...##
###
Linear approximation of simple objects

1994
*
Computational geometry
*

Let S be

doi:10.1016/0925-7721(94)90015-9
fatcat:ku2mdvrmlbg6thxdhrelyur6jm
*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. ...
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