SIAM Journal on Discrete Mathematics· 2026Q1
On the Number of Pairwise Touching Cylinders in \({\mathbb{R}}\)\(^{{d}}\)
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- 2026year
Short summary
Researchers established exponential upper bounds on the number of mutually touching unit cylinders in d-dimensional space (R^d).
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Key points
- Derived exponential upper bounds on mutually touching unit cylinders in R^d.
- The method uses polynomial equalities and inequalities to model cylinder touching.
- Extends previous work which showed 7 cylinders can mutually touch in R^3.
AI-generated from the title and abstract; the full text is not read.
Abstract
Abstract. 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.
The authors' abstract, as published at the source. SIAM Journal on Discrete Mathematics, 2026 · DOI ↗
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Field: Computer Graphics and Computer-Aided Design
Computer Graphics and Computer-Aided DesignComputer Science