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

Charles Audet, Jordan Ninin
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]

Amol Aggarwal
2010 arXiv   pre-print
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  , 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

D. Bhagavathi, S. Olariu, J.L. Schwing, W. Shen, L. Wilson, J. Zhang
1995 Journal of Parallel and Distributed Computing
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.  ...

### Blind approximation of planar convex sets

M. Lindenbaum, A.M. Bruckstein
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]

Pooya Davoodi, Michiel Smid, Freek van Walderveen
2012 Lecture Notes in Computer Science
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.  ...

### The maximum size of a convex polygon in a restricted set of points in the plane

N. Alon, M. Katchalski, W. R. Pulleyblank
1989 Discrete & Computational Geometry
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.  ...

### Computational Geometry Column 34 [article]

Pankaj K. Agarwal, Joseph O'Rourke
1998 arXiv   pre-print
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.  ...

### Multiplicities of interpoint distances in finite planar sets

Paul Erdős, Peter C. Fishburn
1995 Discrete Applied Mathematics
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.  ...

### Minimum Weight Convex Steiner Partitions

2009 Algorithmica
This O(W log n) bound is the best possible due to Eppstein's lower bound on minimum weight Steiner triangulations  .  ...  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

Zvi Drezner, Atsuo Suzuki
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

Charles Audet, Pierre Hansen, Frédéric Messine
2006 Journal of Global Optimization
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  ...

### Interactive 3D distance field computation using linear factorization

Avneesh Sud, Naga Govindaraju, Russell Gayle, Dinesh Manocha
2006 Proceedings of the 2006 symposium on Interactive 3D graphics and games - SI3D '06
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.  ...

### The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees

Kenneth J. Supowit
1983 Journal of the ACM
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).  ...

### A compact piecewise-linear voronoi diagram for convex sites in the plane

M. McAllister, D. Kirkpatrick, J. Snoeyink
1996 Discrete & Computational Geometry
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.  ...

### Linear approximation of simple objects

Jean-Marc Robert, Godfried T. Toussaint
1994 Computational geometry
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.  ...
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