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  • Author or Editor: Imre Bárány x
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On a question of V. I. Arnol'd

Acta Mathematica Hungarica
Volume/Issue:
Volume 137: Issue 1-2
DOI:
https://doi.org/10.1007/s10474-012-0219-2
Author: Imre Bárány

Abstract

We show by a construction that there are at least exp {cV (d−1)/(d+1)} convex lattice polytopes in ℝd of volume V that are different in the sense that none of them can be carried to an other one by a lattice preserving affine transformation.

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A note on the size of the largest ball inside a convex polytope

Periodica Mathematica Hungarica
Volume/Issue:
Volume 51: Issue 2
DOI:
https://doi.org/10.1007/s10998-005-0026-4
Authors: Imre Bárány and Nándor Simányi

Summary  

Let \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} $m>1$ \end{document} be an integer, \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} $B_m$ \end{document} the set of all unit vectors of \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} $\Bbb R^m$ \end{document} pointing in the direction of a nonzero integer vector of the cube \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} $[-1,\,1]^m$ \end{document}. Denote by \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} $s_m$ \end{document} the radius of the largest ball contained in the convex hull of \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} $B_m$ \end{document}. We determine the exact value of \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} $s_m$ \end{document} and obtain the asymptotic equality \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} $s_m\sim\frac{2}{\sqrt{\log m}}$ \end{document}.

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Planar point sets with a small number of empty convex polygons

Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 41: Issue 2
DOI:
https://doi.org/10.1556/sscmath.41.2004.2.4
Authors: Imre Bárány and Pável Valtr

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ˜1.62n 2 empty triangles, ˜1.94n 2 empty quadrilaterals, ˜1.02n 2 empty pentagons, and ˜0.2n 2 empty hexagons.

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Covering lattice points by subspaces

Periodica Mathematica Hungarica
Volume/Issue:
Volume 43: Issue 1-2
DOI:
https://doi.org/10.1023/a:1015233631926
Authors: Imre Bárány, Gergely Harcos, János Pach, and Gábor Tardos
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