DOI: 10.1137/25m1836842 ISSN: 0895-4801
On the Number of Pairwise Touching Cylinders in \({\mathbb{R}}\)\(^{{d}}\)
Jozsef Solymosi, Joshua ZahlAbstract.
John E. Littlewood posed the question “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.” Bozóki, Lee, and Rónyai constructed a configuration of 7 mutually touching unit cylinders. The best-known upper bounds show that at most 10 unit cylinders in [Formula: see text] can mutually touch. We consider this problem in higher dimensions, and obtain exponential (in [Formula: see text]) upper bounds on the number of mutually touching cylinders in [Formula: see text]. 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 nonequalities.