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

TL;DR

This paper advances the problem of how many cylinders can touch each other in $\mathbb{R}^d$ by formulating tangency as polynomial constraints on a line-parameter space and applying real-algebraic-geometric tools. Through a novel use of the restricted Zariski closure and Milnor–Thom type bounds, the authors derive exponential-in-$d$ upper bounds: $|C|\le 4d\cdot 7^{2d-3}$ for unit-radius cylinders and $|C|\le 20(d+1)\cdot 7^{2d-2}$ for arbitrary radii, thereby establishing finiteness for all $d\ge 4$. The method treats cylinder touching as a mix of polynomial equalities and inequalities, enabling a purely algebraic approach to a geometric incidence problem and connecting to the unit-distance line problem in higher dimensions. Overall, the work introduces a flexible polynomial-method framework for higher-dimensional touching configurations with potential applications to related distance-set questions in real algebraic geometry.

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.''} 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 $\mathbb{R}^3$ 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.

Theorems & Definitions (11)

• Theorem 1.1
• Theorem 1.2
• Lemma 3.1
• proof
• Remark 3.1
• Theorem 4.1
• Corollary 4.2
• Theorem 4.3
• Corollary 4.4
• proof : Proof of Theorem \ref{['unitDistanceLines']}