G. Fejes Toth posed the following problem: Determine the infimum of the densities of the lattices of closed balls in $bR^n$ such that each affine $k$-subspace $(0 le k le n-1)$ of $bR^n$ intersects some ball of the lattice. We give a lower estimate for any $n,k$ like above. If, in the problem posed by G. Fejes Toth, we replace the ball $B^n$ by a (centrally symmetric) convex body $Ksubset bR^n$, we may ask for the infimum of all above infima of densities of lattices of translates of $K$ with the above property, when $K$ ranges over all (centrally symmetric) convex bodies in $bR^n$. For these quantities we give lower estimates as well, which are sharp, or almost sharp, for certain classes of convex bodies $K$. For $k=n-1$ we give an upper estimate for the supremum of all above infima of densities, $K$ also ranging as above (i.e., a minimax problem). For $n=2$ our estimate is rather close to the conjecturable maximum. We point out the connection of the above questions to the following problem: Find the largest radius of a cylinder, with base an $(n-1)$-ball, that can be fitted into any lattice packing of balls (actually, here balls can be replaced by some convex bodies $K subset bR^n$, the axis of the cylinder may be $k$-dimensional and its basis has to be chosen suitably). Among others we complete the proof of a theorem of I. Hortobagyi from 1971. Our proofs for the lower estimates of densities for balls, and for the cylinder problem, follow quite closely a paper of J. Horvath from 1970. This paper is also an addendum to a paper of the first named author from 1978 in the sense that to some arguments given there not in a detailed manner, we give here for all of these complete proofs.
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