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 $\pi \mathrm{/}\sqrt{12}$). Moreover, we prove that for every $\epsilon >0$ there exists a nonparallel cylinder packing with lower density greater then $\pi \mathrm{/}6-\epsilon$.