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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On-line covering the unit square with squares

Januszewski J.

Bulletin of the Polish Academy of Sciences. Mathematics

2009

Vol. 57, no 1

57-62

The unit square can be on-line covered with any sequence of squares whose total area is not smaller than 4.

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 .