On-line covering of the unit cube by boxes and by convex bodies

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 .