Density estimates for $k$-impassable lattices of balls and general convex bodies in ${\Bbb R}^n$

TL;DR

This paper investigates the minimal densities of lattice packings of convex bodies in n-dimensional space that intersect all affine k-subspaces, providing sharp bounds, connections to cylinder packing problems, and completing previous proofs.

Contribution

It offers new lower bounds for densities of lattice packings intersecting all affine k-subspaces, extends results to convex bodies, and addresses the cylinder packing problem, building on and completing prior work.

Findings

Lower bounds for lattice densities in various dimensions and convex bodies.

Sharp or near-sharp estimates for specific classes of convex bodies.

Connection established between lattice packings and maximal cylinder radius in packings.

Abstract

G. Fejes T\'oth 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 T\'oth, we replace the ball $B^n$ by a (centrally symmetric) convex body $K\subset \bR^n$, 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. Hortob\'agyi from 1971. Our proofs for the lower estimates of densities for balls, and for the cylinder problem, follow quite closely a paper of J. Horv\'ath 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.