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Convex Polygons in Cartesian Products.
Jean-Lou De Carufel,Adrian Dumitrescu,Wouter Meulemans,Tim Ophelders,Claire Pennarun,Csaba D. Tóth,Sander Verdonschot +6 more
TLDR
It is proved that every such grid contains a convex polygon with Ω(log n) vertices and that this bound is tight up to a constant factor, and a tight lower bound is obtained for the maximum number of points in convex position in a d-dimensional grid.Abstract:
We study several problems concerning convex polygons whose vertices lie in a Cartesian product (for short, grid) of two sets of n real numbers. First, we prove that every such grid contains a convex polygon with $\Omega$(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d $\in$ N), and obtain a tight lower bound of $\Omega$(log d--1 n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the largest convex chain in a grid that contains no two points of the same x-or y-coordinate. We show how to efficiently approximate the maximum size of a supported convex polygon up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors.read more
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Continuous similarity measures for curves and surfaces
Abstract: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.
References
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Book ChapterDOI
A combinatorial problem in geometry
P. Erdös,G. Szekeres +1 more
TL;DR: In this paper, the present problem has been suggested by Miss Esther Klein in connection with the following proposition: "Our present problem is the same problem as the one suggested by the author of this paper."
Proceedings ArticleDOI
Topologically sweeping an arrangement
TL;DR: The advantages of sweeping with a topological line that is not necessarily straight are demonstrated and an arrangement of n lines in the plane can be swept over in O ( n 2 ) time and O(n) space by a such a line.
Journal ArticleDOI
A lower bound for the volume of strictly convex bodies with many boundary lattice points
TL;DR: In this paper, it was shown that the restriction to r,(Q|c'(n)[S(q]"/t""1) is unnecessary and that the above theorem implies that ViQ>k'in)Nin+1)li"-l) where k
) > 0 is a constant depending only on n.
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