- I am a researcher on convexity and discrete geometry.edit
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Research Interests: Mathematics and Radius
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We present an on-line covering method which enables covering the unit cube of the Euclidean d-space by every sequence of boxes of sides at most 1 whose total volume is at least 2x8^{d-1) - 2^{d-1}.
We generalize Monge's theorem for n+1 pairwise homothetic convex bodies in E^n in place of three disks in E^2. We also present a version for homotheties for pairs of vertices of a non degenerate simplex in E^n. It includes a reverse... more
We generalize Monge's theorem for n+1 pairwise homothetic convex bodies in E^n in place of three disks in E^2. We also present a version for homotheties for pairs of vertices of a non degenerate simplex in E^n. It includes a reverse of Monge's theorem. Moreover, we give an analogon of Monge's theorem on the n-dimensional sphere.
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Research Interests: DDC and Axial Symmetry
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Denote by Km the mirror image of a planar convex body K in a straight line m. It is easy to show that K∗ m = conv(K ∪ Km) is the smallest (by inclusion) convex body whose axis of symmetry is m and which contains K. The ratio axs(K) of the... more
Denote by Km the mirror image of a planar convex body K in a straight line m. It is easy to show that K∗ m = conv(K ∪ Km) is the smallest (by inclusion) convex body whose axis of symmetry is m and which contains K. The ratio axs(K) of the area of K to the minimum area of K∗ m is a measure of axial symmetry of K. A question is how to find a line m in order to guarantee that K∗ m be of the smallest possible area. A related task is to estimate axs(K) for the family of all convex bodies K. We give solutions for the classes of triangles, right-angled triangles and acute triangles. In particular, we prove that axs(T ) > 1 2 √ 2 for every triangle T , and that this estimate cannot be improved in general. MSC 2000: 52A10, 52A38
Research Interests: Mathematics and Cube
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We prove that the Banach-Mazur distance between arbitrary two central sections of co-dimension c of any centrally symmetric convex body in E is at most (2c + 1). MSC 2000: 52A21, 46B20
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A convex body R in a normed d-dimensional space M is called reduced if the M-thickness ∆(K) of each convex body K ⊂ R different from R is smaller than ∆(R). We present two characterizations of reduced polytopes in M. One of them is that a... more
A convex body R in a normed d-dimensional space M is called reduced if the M-thickness ∆(K) of each convex body K ⊂ R different from R is smaller than ∆(R). We present two characterizations of reduced polytopes in M. One of them is that a convex polytope P ⊂ M is reduced if and only if through every vertex v of P a hyperplane strictly supporting P passes such that the M-width of P in the perpendicular direction is ∆(P ). Also two characterization of reduced simplices in M and a characterization of reduced polygons in M are given. MSC 2000: 52A21, 52B11, 46B20
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We consider packing a triangle with a number of equal positive homothetical copies. In particular, we show that every triangle can be packed with 7 copies of ratio \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts}... more
We consider packing a triangle with a number of equal positive homothetical copies. In particular, we show that every triangle can be packed with 7 copies of ratio \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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{2}{7}$$ \end{document}, with 8 copies of ratio \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} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{3}{{11}}$$ \end{document}, and with 9 and 10 copies of ratio \docume...
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The aim of this paper is to show an on-line algorithm for packing sequences of d-dimensional boxes of edge lengths at most 1 in a box of edge lengths at least 1. It is more efficient than a previously known algorithm in the case when... more
The aim of this paper is to show an on-line algorithm for packing sequences of d-dimensional boxes of edge lengths at most 1 in a box of edge lengths at least 1. It is more efficient than a previously known algorithm in the case when packing into a box with short edges. In particular, our method permits packing every sequence of boxes of edge lengths at most 1 and of total volume at most <img src="/fulltext-image.asp?format=htmlnonpaginated&src=H2102H44536062L1_html\10998_2004_Article_148134_TeX2GIFIE1.gif" border="0" alt=" $$(1 - \frac{1} {2}\sqrt 3 )^{d - 1} $$ " /> in the unit cube. For packing sequences of convex bodies of diameters at most 1 the result is d! times smaller.