2016/12/05 by Makai,, E., Martini, H.
#FOS: Mathematics #Metric Geometry (math.MG) #Primary: 52C07. Secondary: 52A40
paper · doi:10.48550/arxiv.1612.01307
G. Fejes Tóth posed the following problem: Determine the infimum of the densities of the lattices of closed balls in \bRn such that each affine k-subspace (0 ≤ k ≤ n-1) of \bRn 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 Tóth, we replace the ball Bn by a (centrally symmetric) convex body K⊂ \bRn, 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 \bRn. 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 ⊂ \bRn, 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. Hortobágyi from 1971. Our proofs for the lower estimates of densities for balls, and for the cylinder problem, follow quite closely a paper of J. Horváth 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.