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Content available remote Diameter of reduced spherical convex bodies
Lassak M., Musielak M.
Fasciculi Mathematici
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2018
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Nr 61
103--108
EN
The intersection L of two different non-opposite hemispheres of the unit sphere S2 is called a lune. By Δ(L) we denote the distance of the centers of the semicircles bounding L. By the thickness Δ(C) of a convex body C ⊂ S2 we mean the minimal value of Δ(L) over all lunes L ⊃ C. We call a convex body R ⊂ S2 reduced provided Δ(Z) < Δ(R) for every convex body Z being a proper subset of R. Our aim is to estimate the diameter of R, where Δ(R) < π/2, in terms of its thickness.
2
Content available remote Reduced spherical convex bodies
Lassak M., Musielak M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2018
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Vol. 66, no. 1
87--97
EN
The paper presents a number of properties of reduced bodies on the two-dimensional sphere. Our main theorem describes the shape of reduced bodies of thickness below π/2.
3
Content available remote Banach-Mazur distance between convex quadrangles
Lassak M.
Demonstratio Mathematica
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2014
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Vol. 47, nr 4
989--993
EN
It is proved that the Banach–Mazur distance between arbitrary two convex quadrangles is at most 2. The distance equals 2 if and only if the pair of these quadrangles is a parallelogram and a triangle.
4
Content available remote On-line covering of the unit cube by boxes and by convex bodies
Januszewski J., Lassak M.
Bulletin of the Polish Academy of Sciences. Mathematics
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2003
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Vol. 51, no 3
309-317
EN
The paper presents a method of q-adic on-line covering of the unit d-dimensional cube I[sup]d by arbitrary sequence of boxes of side lengths of the form q[sup]-k for k [is an element of {O, l, 2,...} whose total volume is a number of the order of magnitude 2[sup]d. We also show that every sequence of boxes of side lengths at most 1 and of the total volume at least 4[sup]d. 2.566 ... permits an on-line covering of I[sup]d. Moreover, we estimate the total volume of sequences of convex bodies of diameters at most l which permit an on-line covering of I .
5
Content available remote Zagadnienie Auerbacha-Banacha-Mazura-Ulama pakowania worka ziemniaków
Lassak M.
Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne
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1998
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[Z] 34
49-59
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