|ISTVAN TALATA, Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310, USA|
|On translative coverings of a convex body with its homothetic copies of given total volume|
Let K be a d-dimensional convex body. Denote by h(K) the minimum number of smaller homothetic copies of K which are needed to cover K. Furthermore, denote by hv(K) the smallest real number with the property that every sequence of positive homothetic copies of K with total volume at least hv(K) vol(K) permints a translative covering of K. It is clear that .
It is proved by Rogers  that for any d-dimensional convex body K there exists a covering of Rd with translates of K with density at most . As already Rogers observed in 1967, this result implies that for centrally symmetric convex bodies. Similarly, was proved for arbitrary convex bodies.
In this talk we show that the method used by Rogers can be extended for homothetic copies of K with different coefficients. This way we can improve on the upper bound of Januszewski  proving for centrally symmetric convex bodies, and for arbitrary d-dimensional convex bodies.