Insertion and expansion of cylinders around the unit ball

Ivan Matić; Radoš Radoičić; Andreas Spomer

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arXiv · 2610.13972

Insertion and expansion of cylinders around the unit ball

Ivan Matić·Radoš Radoičić·Andreas Spomer

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Abstract

A unit ball is surrounded with a ring of $n$ equal infinite circular cylinders. Each cylinder touches the ball and its two neighbors. Can one more cylinder of the same radius be added if the configuration may be rearranged? Can the $n$ cylinders instead be enlarged, and by how much? We prove that insertion is impossible for $3\le n\le7$ and possible for $n\ge9$. The case $n=8$ remains open. The case $n=6$ is Kuperberg's problem from 1990, which was resolved recently. We prove that the ring of four cylinders cannot be expanded. This settles a conjecture of Ogievetsky and Shlosman from 2018. For $n\ge5$ the largest possible expansion radius remains unknown.

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Ivan Matić, Radoš Radoičić, Andreas Spomer. 2026-10-14. Insertion and expansion of cylinders around the unit ball. https://arxiv.org/abs/2610.13972

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