On a problem of G. Fejes Tóth (Q1322366)

The authors prove the following Theorem: Let \(L \subset E^ 3\) be a lattice and \(B^ 3\) the unit 3-ball. If each straight line meets \(L+B^ 3\), then \(\text{det} L \leq 2 (4/3)^ 3\). In terms of density such an arrangement has a density \(V(B^ 3)/ \text{det} L = 0,8835\dots\) The theorem is a partial solution of the following general problem by G. Fejes Tóth: In \(n\)-space find the thinnest lattice arrangement of closed balls such that every \(k\)-dimensional \((0 \leq k \leq n - 1)\) flat meets one of these balls.