Covering a set with homothets of a convex body

We consider two problems mentioned in the book “Research Problems in Discrete Geometry” (Brass et al. in research problems in discrete geometry, vol xii+499. Springer, New York, pp ISBN: 978-0387-23815-8; 0-387-23815-8, 2005). First, let K and L be given convex bodies in mathbbR^d{mathbb{R}^{d}} . We prove that if the total volume of a family of positive homothets of K is sufficiently large then they permit a translative covering of L. This problem, in the case when K = L and the dimension is two, was originally posed by L. Fejes Tóth. The previously known bound (Januszewski in proc. of the International scientific conference on mathematics, pp 29–34. Žilina, 1998) on the total volume (in the case when K = L) was of order d ^d vol(K), we prove a bound that is exponential in the dimension. The second problem is the following: Find a condition, in terms of the coefficients of homothety, that is necessary for a family of positive homothets of K to cover K. The problem was phrased by V. Soltan, who conjectured that the sum of the coefficients is at least d. We confirm an asymptotic version of this conjecture.