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Banach-Mazur distance between convex quadrangles

Lassak M.

Demonstratio Mathematica

2014

Vol. 47, nr 4

989--993

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.

Covering a reduced polygon by a disk

Fabińska E.

Demonstratio Mathematica

2009

Vol. 42, nr 4

851-855

A convex body R in Euclidean d-space Ed is reduced if every convex body K C R different from R has thickness smaller than the thickness A(R) of R. We prove that every reduced polygon P C E2 is contained in a disk of radius A(P) centered at a boundary point of P.

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

2003

Vol. 51, no 3

309-317

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 .

Zagadnienie Auerbacha-Banacha-Mazura-Ulama pakowania worka ziemniaków

Lassak M.

Roczniki Polskiego Towarzystwa Matematycznego. Seria 2: Wiadomości Matematyczne

1998

[Z] 34

49-59