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  • Author or Editor: Károly Bezdek x
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Summary  

The Illumination Conjecture was raised independently by Boltyanski and Hadwiger in 1960. According to this conjecture any \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} $d$ \end{document}-dimensional convex body can be illuminated by at most \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} $2^d$ \end{document} light sources. This is an important fundamental problem. The paper surveys the state of the art of the Illumination Conjecture.

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Abstract  

Let S d-1 denote the (d − 1)-dimensional unit sphere centered at the origin of the d-dimensional Euclidean space. Let 0 < α < π. A set P of points in S d-1 is called almost α-equidistant if among any three points of P there is at least one pair lying at spherical distance α. In this note we prove upper bounds on the cardinality of P depending only on d.

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Abstract  

Let 0 < c < s be fixed real numbers such that

\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} $${c \mathord{\left/ {\vphantom {c s}} \right. \kern-\nulldelimiterspace} s} \leqslant {{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$ \end{document}
, and let f : E2 → Ed for d ≥ 2 be a function such that for every p, qE 2 if p − q = c, then f(p) − f(q) ≤ c, and if p − q = s, then f(p) − f(q) ≥ s. Then f is a congruence. This result depends on and expands a result of Rdo et. al. [9], where a similar result holds, but for
\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} $${{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } 3}} \right. \kern-\nulldelimiterspace} 3}$$ \end{document}
replacing
\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} $${{\left( {\sqrt 5 - 1} \right)} \mathord{\left/ {\vphantom {{\left( {\sqrt 5 - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}$$ \end{document}
. We also present a further extensions where E2 is replaced by En for n > 2 and where the range of c/s is enlarged.

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Abstract  

In this paper we prove some stronger versions of Danzer-Grnbaum's theorem including the following stability-type result. For 0 < α < 14π/27 the maximum number of vertices of a convex polyhedron in E 3 such that all angles between adjacent edges are bounded from above by α is 8. One of the main tools is the spherical geometry version of Pl's theorem.

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Abstract  

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in

\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} $$\mathbb{E}^d$$ \end{document}
, which are centers of balls Bd(pi, r i) and Bd(qi, r i) of radius r i, for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over
\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} $$\mathbb{E}^d$$ \end{document}
, where the functions in question are constructed from maximum and minimum operations applied to functions each being radially symmetric monotone decreasing and integrable.

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Summary  

The notion of successive illumination parameters of convex bodies is introduced. We prove some theorems in the plane and determine the exact values of the successive illumination parameters of spheres, cubes and cross-polytopes for some dimensions.

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