On a relation between packing and covering densities of convex bodies (Q2022613)

A conjecture attributed to \textit{W. Kuperberg} [Geom. Dedicata 23, 59--66 (1987; Zbl 0613.52007)] says the following: Let \(C\) be a convex body with interior points in \({\mathbb R}^d\), \(d\ge 2\). For any \(\varepsilon>0\) there exists \(\delta>0\) such that the covering density \(\delta(C)\) and the packing density \(\theta(C)\) (both in the translative sense) satisfy \(\delta(C)\le 1-\varepsilon\Rightarrow \theta(C)\ge 1+\delta\) and \(\theta(C)\ge 1+\varepsilon\Rightarrow \delta(C)\le 1-\delta\). This conjecture is proved here in a refined form. The proof is based on estimating the number of steps of a certain greedy algorithm.