On non-parallel cylinder packings

In this paper we will discuss optimal lower and upper density of non-parallel cylinder packings in $R^{3}$ and similar problems. The main result of the paper is a proof of the conjecture of K. Kuperberg for upper density (existence of a non-parallel cylinder packing with upper density ${\pi}/{\sqrt{12}}$). Moreover, we prove that for every $\varepsilon > 0$ there exists a nonparallel cylinder packing with lower density greater then ${\pi}/{6} - \varepsilon$.