منابع مشابه
Covering space with convex bodies
1. A few years ago Rogers [1] showed that, if K is any convex body in n-dimensional Euclidian space, there is a covering of the whole space by translates of K with density less than nlogn+nloglogn+5n, provided n > 3. However the fact that the covering density is reasonably small does not imply that the maximum multiplicity is also small. In the natural covering of space by closed cubes, the den...
متن کاملThe Covering Index of Convex Bodies
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel [Mathematika 52 (2005), 47–52] introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce a relative of the covering parameter called covering index, which turn...
متن کاملOn the covering index of convex bodies
Covering a convex body by its homothets is a classical notion in discrete geometry that has resulted in a number of interesting and long-standing problems. Swanepoel introduced the covering parameter of a convex body as a means of quantifying its covering properties. In this paper, we introduce two relatives of the covering parameter called covering index and weak covering index, which upper bo...
متن کاملA Note on Covering by Convex Bodies
A classical theorem of Rogers states that for any convex body K in n-dimensional Euclidean space there exists a covering of the space by translates of K with density not exceeding n log n+n log log n+5n. Rogers’ theorem does not say anything about the structure of such a covering. We show that for sufficiently large values of n the same bound can be attained by a covering which is the union of ...
متن کاملCovering convex bodies by cylinders and lattice points by flats ∗
In connection with an unsolved problem of Bang (1951) we give a lower bound for the sum of the base volumes of cylinders covering a d-dimensional convex body in terms of the relevant basic measures of the given convex body. As an application we establish lower bounds on the number of k-dimensional flats (i.e. translates of k-dimensional linear subspaces) needed to cover all the integer points o...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1962
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-7-3-281-285