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. Hungar. 39 153 159 Nyklová, H., Empty and almost empty polygons (Master Thesis), Charles University, Prague (1999

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-szekeres convex polygon theorem, lect. notes comput. sci. 2098 (2001), 91-105. Problems and results around the erdös-szekeres convex polygon theorem Lect. notes comput. sci

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Abstract  

We prove that the interior of every convex polygon with n vertices (n ≥ 4) can be illuminated by four 45°-vertex lights. We restrict each vertex to anchoring at most one floodlight. This answers a question of O’Rourke, Shermer and Streinu [5].

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DOBKIN, D., EDELSBRUNNER, H. and OVERMARS, M., Searching for empty convex polygons, Algorithmica 5 (1990), 561-571. MR 91g :68160 Searching for empty convex polygons 5

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A bicycle ( n , k )-gon is an equilateral n -gon whose k -diagonals are equal. S. Tabach-nikov proved that a regular n -gon is first-order flexible as a bicycle ( n , k )-gon if and only if there is an integer 2 ≦ rn -2 such that tan (π/ n ) tan ( kr π/ n ) = tan ( k π/ n ) tan ( r π/ n ). In the present paper, we solve this trigonometric diophantine equation. In particular, we describe the family of first order flexible regular bicycle polygons.

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Answering a question of H. Harborth, for any given a 1,...,a n > 0, satisfying

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$a_i < \sum\limits_{j \ne i} {a_j }$$ \end{document}
we determine the infimum of the areas of the simple n-gons in the Euclidean plane, having sides of length a 1,...,a n (in some order). The infimum is attained (in limit) if the polygon degenerates into a certain kind of triangle, plus some parts of zero area. We show the same result for simple polygons on the sphere (of not too great length), and for simple polygons in the hyperbolic plane. Replacing simple n-gons by convex ones, we answer the analogous questions. The infimum is attained also here for degeneration into a certain kind of triangle.

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A methodology of comparison is presented as a way to determine the origin of ceramic fragments proceeding from clays and archaeological sites of the 9th Region of Chile, analyzed through X-ray fluorescence (XRF) spectroscopy. Net areas of peak of sample spectra, are analysed using the graphic polygonal method, developed previously.1,2 Criterions of comparison have been defined as a way to determine similarity degrees and proceeding places. Polygonal representations used are triangles, squares, pentagons and hexagons; the similarity grading among samples is defined. Results are compared with clay samples and sand removing effects on fragments are also defined.

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. , Planar point sets with a small number of empty convex polygons, Stud. Sci. Math. Hung. , 41 (2004), 243–266. Valtr P. Planar point sets with a small number of empty convex polygons

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Hosono, K. and Urabe, M. , On the minimum size of a point set containing two non-intersecting empty convex polygons, JCDCG 2004, LNCS , 3742 (2005), 117

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