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arXiv · 2512.24595

On the number of pairwise touching cylinders in $\mathbb{R}^d$

Jozsef Solymosi·Josh Zahl
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Abstract

John E. Littlewood posted the question {\em ``Is it possible in 3-space for seven infinite circular cylinders of unit radius each to touch all the others? Seven is the number suggested by counting constants.''} Boz\'oki, Lee, and R\'onyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in $\mathbb{R}^3$ can mutually touch. We consider this problem in higher dimensions, and obtain exponential (in $d$) upper bounds on the number of mutually touching cylinders in $\mathbb{R}^d$. Our method is fairly flexible, and it makes use of the fact that cylinder touching can be expressed as a combination of polynomial equalities and non-equalities.

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Jozsef Solymosi, Josh Zahl. 2025-12-31. On the number of pairwise touching cylinders in $\mathbb{R}^d$. https://arxiv.org/abs/2512.24595

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