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1
Banach-Mazur distance between convex quadrangles
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.
2
Some results about +nieskończoność-, -nieskończoność- and i-chord functions
EN
This paper concerns determination of a convex bodies by values of +oo- -oo-and i-chord functions. We prove that any at least two-dimensional convex body is not determined by values of -oo-chord functions at any two internal points. We also present some positive results on determination of convex bodies using -oo- or +oo-chord function at one point and i-chord function at other one.
3
On-line covering of the unit cube by boxes and by convex bodles
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 .
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