## 1 Introduction

Open and closed Euclidean unit n-balls will be denoted by $$\mathbb {B}^n$$ and $$\mathbb {D}^n$$ respectively. The closed unit interval is denoted by $$\mathbb {I}$$. A general polycylinder C is a set congruent to $$\prod _{i=1}^{i=m}\lambda _i\mathbb {D}^{k_i}$$ in $$\mathbb {R}^{ k_1+\dots + k_m}$$, where $$\lambda _i$$ is in $$[0,\infty ]$$. 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 $$\mathbb {D}^2 \times \mathbb {R}^n$$ in $$\mathbb {R}^{n+2}$$. A polycylinder packing of $$\mathbb {R}^{n+2}$$ is a family $$\mathscr {C} = \{C_i\}_{i \in I}$$ of polycylinders $$C_i \subset \mathbb {R}^{n+2}$$ with mutually disjoint interiors. The upper density $$\delta ^+ (\mathscr {C})$$ of a packing $$\mathscr {C}$$ of $$\mathbb {R}^n$$ is defined to be

\begin{aligned} \delta ^+ (\mathscr {C}) = \limsup _{r\rightarrow \infty }\tfrac{ {{\mathrm{Vol}}}(\mathscr {C}\cap r\mathbb {B}^n)}{{{\mathrm{Vol}}}(r\mathbb {B}^n)}. \end{aligned}

The upper packing density $$\delta ^+ (C)$$ of an object C is the supremum of $$\delta ^+ (\mathscr {C})$$ over all packings $$\mathscr {C}$$ of $$\mathbb {R}^n$$ by C.

This article proves the following sharp bound for the packing density of infinite polycylinders:

### Theorem 1

$$\delta ^+(\mathbb {D}^2 \times \mathbb {R}^n) = \pi /\sqrt{12}$$ 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.

## 2 Transversality

This section introduces the required transversality arguments from affine geometry. A d-flat is a d-dimensional affine subspace of $$\mathbb {R}^n$$. The parallel dimension $$\mathrm{dim}_\parallel \{F,\dots , G\}$$ of a collection of flats $$\{F, \dots , G\}$$ is the dimension of their maximal parallel sub-flats. The notion of parallel dimension can be interpreted in several ways, allowing a modest abuse of notation.

• For a collection of flats $$\{F, \dots , G\}$$, consider their tangent cones at infinity $$\{F_\infty , \dots , G_\infty \}$$. The parallel dimension of $$\{F, \dots , G\}$$ is the dimension of the intersection of these tangent cones. This may be viewed as the limit of a rescaling process $$\mathbb {R}^n \rightarrow r\mathbb {R}^n$$ as r tends to 0, leaving only the scale-invariant information.

• For a collection of flats $$\{F, \dots , G\}$$, consider each flat as a system of linear equations. The corresponding homogeneous equations determine a collection of linear subspaces $$\{F_\infty , \dots , G_\infty \}$$. The parallel dimension is the dimension of their intersection $$F_\infty \,\cap \dots \cap \, G_\infty$$.

Two disjoint d-flats are parallel if their parallel dimension is d, that is, if every line in one is parallel to a line in the other.

### Lemma 1

A pair of disjoint n-flats in $$\mathbb {R}^{n+k}$$ with $$n\ge k$$, has parallel dimension strictly greater than $$n-k.$$

### Proof

Let F and G be such a pair. By homogeneity of $$\mathbb {R}^{n+k}$$, let $$F=F_\infty .$$ As $$F_\infty$$ and G are disjoint, G contains a non-trivial vector $$\mathbf{v}$$ such that $$G = G_\infty +\mathbf{v}$$ and $$\mathbf{v}$$ is not in $$F_\infty + G_\infty .$$ It follows that

\begin{aligned} \mathrm{dim} (\mathbb {R}^{n+k})\ge & {} \mathrm{dim}\big (F_\infty + G_\infty + {{\mathrm{span}}}(\mathbf{v})\big )> \mathrm{dim}(F_\infty + G_\infty )\\= & {} \mathrm{dim} (F_\infty ) + \mathrm{dim} (G_\infty ) - \mathrm{dim} (F_\infty \cap G_\infty ). \end{aligned}

Count dimensions to find $$n+k > n +n - \mathrm{dim}_\parallel (F_\infty ,G_\infty ).$$ $$\square$$

### Corollary 1

A pair of disjoint n-flats in $$\mathbb {R}^{n+2}$$ has parallel dimension at least $$n-1$$.

## 3 Dimension Reduction

### 3.1 Pairwise Foliations

The core $$a_i$$ of a polycylinder $$C_i$$ congruent to $$\mathbb {D}^2 \times \mathbb {R}^n$$ in $$\mathbb {R}^{n+2}$$ is the distinguished n-flat defining $$C_i$$ as the set of points at most distance 1 from $$a_i$$. In a packing $$\mathscr {C}$$ of $$\mathbb {R}^{n+2}$$ by polycylinders, Corollary 1 shows that, for every pair of polycylinders $$C_i$$ and $$C_j$$, one can choose parallel $$(n-1)$$-dimensional subflats $$b_i \subset a_i$$ and $$b_j \subset a_j$$ and define a product foliation

\begin{aligned} \mathscr {F}^{b_i,b_j}:\mathbb {R}^{n+2} \rightarrow \mathbb {R}^{n-1} \times \mathbb {R}^3 \end{aligned}

with $$\mathbb {R}^3$$ leaves that are orthogonal to $$b_i$$ and to $$b_j$$. Given a point x in $$a_i$$, there is a distinguished $$\mathbb {R}^3$$ leaf $$F_x^{b_i,b_j}$$ that contains the point x. The foliation $$\mathscr {F}^{b_i,b_j}$$ restricts to foliations of $$C_i$$ and $$C_j$$ with right-circular-cylinder leaves.

### 3.2 The Dirichlet Slice

In a packing $$\mathscr {C}$$ of $$\mathbb {R}^{n+2}$$ by polycylinders, the Dirichlet cell $$D_i$$ associated with a polycylinder $$C_i$$ is the set of points in $$\mathbb {R}^{n+2}$$ which lie no further from $$C_i$$ than from any other polycylinder in $$\mathscr {C}$$. The Dirichlet cells of a packing partition $$\mathbb {R}^{n+2},$$ as $$C_i \subset D_i$$ for all polycylinders $$C_i$$. To bound the density $$\delta ^+(\mathscr {C})$$, 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 $$\mathscr {C}$$ of $$\mathbb {R}^{n+2}$$ 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 $$p_x.$$

Note that $$p_x$$ is a sub-flat of $$F_x^{b_i,b_j}$$ for all j in I.

### 3.3 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

\begin{aligned} d_x = \big \{\bigcap _{j\in I} d^j \big \}. \end{aligned}

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 $$\mathscr {F}^{b_i,b_j}$$ 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 $$C_i \cap F_x^{b_i,b_j}$$ of $$\mathscr {F}^{b_i,b_j}|_{C_i}$$ and the leaf $$C_j \cap F_x^{b_i,b_j}$$ of $$\mathscr {F}^{b_i,b_j}|_{C_j}$$. This reduces the analysis to the case of a pair of cylinders in $$\mathbb {R}^3.$$ 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. $$\square$$

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 $$2/\sqrt{3} -1.$$ $$\square$$

### Lemma 4

Let y and z be points on the circle $$S_x(2/\sqrt{3})$$. If each of y and z is equidistant from $$C_i$$ and $$C_j$$, then the angle yxz is smaller than or equal to $$2\arccos (\sqrt{3} -1) = 85.8828\dots ^\circ .$$

### Proof

Following [1, 4], the existence of a supporting hyperplane of $$C_i$$ that separates $${{\mathrm{int}}}(C_i)$$ from $${{\mathrm{int}}}(C_j)$$ suffices. $$\square$$

In [1], it is shown that planar objects satisfying Lemmas 2, 3 and 4 have area no less than $$\sqrt{12}.$$ As the bound holds for all Dirichlet slices, it follows that $$\delta ^+(\mathbb {D}^2\times \mathbb {R}^n) \le \pi /\sqrt{12}$$ in $$\mathbb {R}^{n+2}.$$ The product of the dense disk packing in the plane with $$\mathbb {R}^n$$ gives a polycylinder packing in $$\mathbb {R}^{n+2}$$ 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.