On the Densest Packing of Polycylinders in Any Dimension

Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {D}^2\times \mathbb {R}^n$$\end{document}D2×Rn equals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi /\sqrt{12}$$\end{document}π/12 for all natural numbers n.


Introduction
Open and closed Euclidean unit n-balls will be denoted by B n and D n respectively. The closed unit interval is denoted by I. A general polycylinder C is a set congruent to i=m i=1 λ i D k i in R k 1 +···+k m , where λ i is in [0, ∞]. For this article, the term polycylinder refers to the special case of an infinite polycylinder over a two-dimensional disk of unit radius. A polycylinder is a set congruent to D 2 × R n in R n+2 . A polycylinder packing of R n+2 is a family C = {C i } i∈I of polycylinders C i ⊂ R n+2 with mutually disjoint interiors. The upper density δ + (C ) of a packing C of R n is defined to be The upper packing density δ + (C) of an object C is the supremum of δ + (C ) over all packings C of R n by C. This article proves the following sharp bound for the packing density of infinite polycylinders: 12 for all natural numbers n.
Theorem 1 generalizes a result of Bezdek and Kuperberg [1] and improves on results that may be computed using a method of Fejes Tóth and Kuperberg [3], cf. [2,5]; it gives some of the first sharp upper bounds for packing density in high dimensions.

Transversality
This section introduces the required transversality arguments from affine geometry. Lemma 1 A pair of disjoint n-flats in R n+k with n ≥ k, has parallel dimension strictly greater than n − k.
Proof Let F and G be such a pair. By homogeneity of R n+k , let F = F ∞ . As F ∞ and G are disjoint, G contains a non-trivial vector v such that Corollary 1 A pair of disjoint n-flats in R n+2 has parallel dimension at least n − 1.

Pairwise Foliations
The core a i of a polycylinder C i congruent to D 2 × R n in R n+2 is the distinguished n-flat defining C i as the set of points at most distance 1 from a i . In a packing C of R n+2 by polycylinders, Corollary 1 shows that, for every pair of polycylinders C i and C j , one can choose parallel (n − 1)-dimensional subflats b i ⊂ a i and b j ⊂ a j and define a product foliation with R 3 leaves that are orthogonal to b i and to b j . Given a point x in a i , there is a that contains the point x. The foliation F b i ,b j restricts to foliations of C i and C j with right-circular-cylinder leaves.

The Dirichlet Slice
In a packing C of R n+2 by polycylinders, the Dirichlet cell D i associated with a polycylinder C i is the set of points in R n+2 which lie no further from C i than from any other polycylinder in C . The Dirichlet cells of a packing partition R n+2 , as C i ⊂ D i for all polycylinders C i . To bound the density δ + (C ), it is enough to fix an i in I and consider the density of C i in D i .
Consider the following slicing of the Dirichlet cell D i . Given a fixed polycylinder C i in a packing C of R n+2 by polycylinders and a point x on the core a i , the plane p x is the 2-flat orthogonal to a i and containing the point x. The Dirichlet slice d x is the intersection of D i and p x .

Bezdek-Kuperberg Bound
For any point x on the core a i of a polycylinder C i , the results of Bezdek and Kuperberg [1] apply to the Dirichlet slice d x .

Lemma 2 A Dirichlet slice is convex and, if bounded, a parabola-sided polygon.
Proof Construct the Dirichlet slice d x as an intersection. Define d j to be the set of points in p x which lie no further from C i than from C j . Then the Dirichlet slice d x is realized as Each arc of the boundary of d x in p x is given by an arc of the boundary of some d j in p x . The boundary of d j in p x is the set of points in p x equidistant from C i and C j . Since the foliation F b i ,b j is a product foliation, the arc of the boundary of d j in p x is also the set of points in p x equidistant from the leaf This reduces the analysis to the case of a pair of cylinders in R 3 . From [1], it follows that d j is convex and the boundary of d j in p x is a parabola; the intersection of such sets d j in p x is convex, and a parabola-sided polygon if bounded.
Let S x (r ) be the circle of radius r in p x centered at x.

Lemma 3 The vertices of d x are not closer to S x (1) than the vertices of a regular hexagon circumscribed about S x (1).
Proof A vertex of d x occurs where three or more polycylinders are equidistant, so the vertex is the center of a (n + 2)-ball B tangent to three polycylinders. Thus B is tangent to three disjoint unit (n + 2)-balls B 1 , B 2 , B 3 . By projecting into the affine hull of the centers of B 1 , B 2 , B 3 , it is immediate that the radius of B is no less than 2/ √ 3 − 1.

Lemma 4
Let y and z be points on the circle S x (2/ √ 3). If each of y and z is equidistant from C i and C j , then the angle yx z is smaller than or equal to 2 arccos( Proof Following [1,4], the existence of a supporting hyperplane of C i that separates int(C i ) from int(C j ) suffices.
In [1], it is shown that planar objects satisfying Lemmas 2, 3 and 4 have area no less than √ 12. As the bound holds for all Dirichlet slices, it follows that δ + (D 2 × R n ) ≤ π/ √ 12 in R n+2 . The product of the dense disk packing in the plane with R n gives a polycylinder packing in R n+2 that achieves this density. Combining this with the result of Thue [6] for n = 0 and the result of Bezdek and Kuperberg [1] for n = 1, Theorem 1 follows.