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• Author or Editor: K. Böröczky
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# Über die Newtonsche Zahl regulärer Vielecke

Periodica Mathematica Hungarica
Author: K. Böröczky
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# Packing of spheres in spaces of constant curvature

Acta Mathematica Hungarica
Author: K. Böröczky
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# Research problems

Periodica Mathematica Hungarica
Authors: A. Hajnal and K. Böröczky
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# Über die dichteste Kugelpackung im hyperbolischen Raum

Acta Mathematica Hungarica
Authors: K. Böröczky and A. Florian
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# THE MINIMUM AREA OF A SIMPLE POLYGON WITH GIVEN SIDE LENGTHS

Periodica Mathematica Hungarica
Authors: K. Böröczky, G. Kertész, and E. Makai

## Abstract

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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# On the volume product of planar polar convex bodies — Lower estimates with stability

Studia Scientiarum Mathematicarum Hungarica
Authors: K. Böröczky, E. Makai, M. Meyer, and S. Reisner

Let K ⊂ ℝ2 be an o-symmetric convex body, and K* its polar body. Then we have |K| · |K*| ≧ 8, with equality if and only if K is a parallelogram. (|·| denotes volume). If K ⊂ ℝ2 is a convex body, with o ∈ int K, then |K| · |K*| ≧ 27/4, with equality if and only if K is a triangle and o is its centroid. If K ⊂ ℝ2 is a convex body, then we have |K| · |[(KK)/2)]*| ≧ 6, with equality if and only if K is a triangle. These theorems are due to Mahler and Reisner, Mahler and Meyer, and to Eggleston, respectively. We show an analogous theorem: if K has n-fold rotational symmetry about o, then |K| · |K*| ≧ n 2 sin2(π/n), with equality if and only if K is a regular n-gon of centre o. We will also give stability variants of these four inequalities, both for the body, and for the centre of polarity. For this we use the Banach-Mazur distance (from parallelograms, or triangles), or its analogue with similar copies rather than affine transforms (from regular n-gons), respectively. The stability variants are sharp, up to constant factors. We extend the inequality |K| · |K*| ≧ n 2 sin2(π/n) to bodies with o ∈ int K, which contain, and are contained in, two regular n-gons, the vertices of the contained n-gon being incident to the sides of the containing n-gon. Our key lemma is a stability estimate for the area product of two sectors of convex bodies polar to each other. To several of our statements we give several proofs; in particular, we give a new proof for the theorem of Mahler-Reisner.

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