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On non-parallel cylinder packings

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arxiv 2310.06943 v1 pith:L2E5LRQM submitted 2023-10-10 math.MG

classification math.MG
keywords cylinderdensitynon-parallelupperlowerpackingpackingsvarepsilon
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abstract

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 ${\pi}/{6} - \varepsilon$.

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