Equal Area Polygons in Convex Bodies
نویسندگان
چکیده
In this paper, we consider the problem of packing two or more equal area polygons with disjoint interiors into a convex body K in E such that each of them has at most a given number of sides. We show that for a convex quadrilateral K of area 1, there exist n internally disjoint triangles of equal area such that the sum of their areas is at least 4n 4n+1 . We also prove results for other types of convex polygons K. Furthermore we show that in any centrally symmetric convex body K of area 1, we can place two internally disjoint n-gons of equal area such that the sum of their areas is at least n−1 π sin π n−1 . We conjecture that this result is true for any convex bodies.
منابع مشابه
Affinely Regular Polygons as Extremals of Area Functionals
For any convex n-gon P we consider the polygons obtained dropping a vertex or an edge of P . The area distance of P to such (n − 1)-gons, divided by the area of P , is an affinely invariant functional on n-gons whose maximizers coincide with the affinely regular polygons. We provide a complete proof of this result. We extend these area functionals to planar convex bodies and we present connecti...
متن کاملOn the variance of random polygons
A random polygon is the convex hull of uniformly distributed random points in a convex body K ⊂ R. General upper bounds are established for the variance of the area of a random polygon and also for the variance of its number of vertices. The upper bounds have the same order of magnitude as the known lower bounds on variance for these functionals. The results imply a strong law of large numbers ...
متن کاملOptimal space coverage with white convex polygons
Assume that we are given a set of points some of which are black and the rest are white. The goal is to find a set of convex polygons with maximum total area that cover all white points and exclude all black points. We study the problem on three different settings (based on overlapping between different convex polygons): (1) In case convex polygons are permitted to have common area, we present ...
متن کاملAlgorithm for finding the largest inscribed rectangle in polygon
In many industrial and non-industrial applications, it is necessary to identify the largest inscribed rectangle in a certain shape. The problem is studied for convex and non-convex polygons. Another criterion is the direction of the rectangle: axis aligned or general. In this paper a heuristic algorithm is presented for finding the largest axis aligned inscribed rectangle in a general polygon. ...
متن کاملOverlapping Area Computation between Irregular Polygons for Its Evolutionary Layout Based on Convex Decomposition
Low efficiency of interference calculation has become the bottleneck that restricts further development of the performance of evolutionary algorithm for the polygon layout. To solve the problem, in this paper, we propose an algorithm of calculating overlapping area between two irregular polygons. For this algorithm, at first, two irregular polygons are respectively decomposed into the minimum n...
متن کامل