#### On the Densest Packing of Polycylinders in Any Dimension

Discrete Comput Geom
On the Densest Packing of Polycylinders in Any Dimension
Wöden Kusner 0
Polycylinders 0
Packing 0
Density 0
Slicing 0
Editor in Charge: János Pach
0 Institute of Analysis and Number Theory, Graz University of Technology, Steyrergasse 30/II , 8010 Graz , Austria
Using transversality and a dimension reduction argument, a result of Bezdek and Kuperberg is applied to polycylinders, showing that the optimal packing density of D2 × Rn equals π/√12 for all natural numbers n. Open and closed Euclidean unit n-balls will be denoted by Bn and Dn respectively. The closed unit interval is denoted by I. A general polycylinder C is a set congruent to ii==1m λi Dki in Rk1+···+km , where λi 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 D2 × Rn in Rn+2. A polycylinder packing of Rn+2 is a family C = {Ci }i∈I of polycylinders Ci ⊂ Rn+2 with mutually disjoint interiors. The upper density δ+(C ) of a packing C of Rn is defined to be
1 Introduction
δ+(C ) = lim sup VoVlo(Cl(r∩BrnB)n ) .
r →∞
The upper packing density δ+(C ) of an object C is the supremum of δ+(C ) over all
packings C of Rn by C .
This article proves the following sharp bound for the packing density of infinite
polycylinders:
Theorem 1 δ+(D2 × Rn) = π/√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 Rn. The parallel dimension
dim {F, . . . , G} of a collection of flats {F, . . . , 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, . . . , G}, consider their tangent cones at infinity
{F∞, . . . , G∞}. The parallel dimension of {F, . . . , G} is the dimension of the
intersection of these tangent cones. This may be viewed as the limit of a rescaling
process Rn → r Rn as r tends to 0, leaving only the scale-invariant information.
– For a collection of flats {F, . . . , G}, consider each flat as a system of linear
equations. The corresponding homogeneous equations determine a collection of linear
subspaces {F∞, . . . , G∞}. The parallel dimension is the dimension of their
intersection F∞ ∩ · · · ∩ G∞.
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 Rn+k with n ≥ k, has parallel dimension strictly
greater than n − k.
Proof Let F and G be such a pair. By homogeneity of Rn+k , let F = F∞. As F∞
and G are disjoint, G contains a non-trivial vector v such that G = G∞ + v and v is
not in F∞ + G∞. It follows that
dim(Rn+k ) ≥ dim F∞ + G∞ + span(v) > dim(F∞ + G∞)
= dim(F∞) + dim(G∞) − dim(F∞ ∩ G∞).
Count dimensions to find n + k > n + n − dim (F∞, G∞).
Corollary 1 A pair of disjoint n-flats in Rn+2 has parallel dimension at least n − 1.
3 Dimension Reduction
3.1 Pairwise Foliations
The core ai of a polycylinder Ci congruent to D2 × Rn in Rn+2 is the distinguished
n-flat defining Ci as the set of points at most distance 1 from ai . In a packing C of
Rn+2 by polycylinders, Corollary 1 shows that, for every pair of polycylinders Ci and
C j , one can choose parallel (n − 1)-dimensional subflats bi ⊂ ai and b j ⊂ a j and
define a product foliation
F bi ,b j : Rn+2 →
Rn−1 × R3
with R3 leaves that are orthogonal to bi and to b j . Given a point x in ai , there is a
distinguished R3 leaf Fxbi ,b j that contains the point x . The foliation F bi ,b j restricts to
foliations of Ci and C j with right-circular-cylinder leaves.
3.2 The Dirichlet Slice
In a packing C of Rn+2 by polycylinders, the Dirichlet cell Di associated with a
polycylinder Ci is the set of points in Rn+2 which lie no further from Ci than from any
other polycylinder in C . The Dirichlet cells of a packing partition Rn+2, as Ci ⊂ Di
for all polycylinders Ci . To bound the density δ+(C ), it is enough to fix an i in I and
consider the density of Ci in Di .
Consider the following slicing of the Dirichlet cell Di . Given a fixed polycylinder
Ci in a packing C of Rn+2 by polycylinders and a point x on the core ai , the plane px
is the 2-flat orthogonal to ai and containing the point x . The Dirichlet slice dx is the
intersection of Di and px .
Note that px is a sub-flat of Fxbi ,b j for all j in I.
3.3 Bezdek–Kuperberg Bound
For any point x on the core ai of a polycylinder Ci , the results of Bezdek and Kuperberg
[
1
] apply to the Dirichlet slice dx .
Lemma 2 A Dirichlet slice is convex and, if bounded, a parabola-sided polygon.
Proof Construct the Dirichlet slice dx as an intersection. Define d j to be the set of
points in px which lie no further from Ci than from C j . Then the Dirichlet slice dx is
realized as
dx =
j∈I
d j .
Each arc of the boundary of dx in px is given by an arc of the boundary of some
d j in px . The boundary of d j in px is the set of points in px equidistant from Ci and
C j . Since the foliation F bi ,b j is a product foliation, the arc of the boundary of d j in
px is also the set of points in px equidistant from the leaf Ci ∩ Fxbi ,b j of F bi ,b j |Ci
and the leaf C j ∩ Fxbi ,b j of F bi ,b j |C j . This reduces the analysis to the case of a pair
of cylinders in R3. From [
1
], it follows that d j is convex and the boundary of d j in
px is a parabola; the intersection of such sets d j in px is convex, and a parabola-sided
polygon if bounded.
Let Sx (r ) be the circle of radius r in px centered at x .
Lemma 3 The vertices of dx are not closer to Sx (1) than the vertices of a regular
hexagon circumscribed about Sx (1).
3 − 1.
Proof A vertex of dx 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 B1, B2, B3. By projecting into the affine
hull of the centers of B1, B2, B3, it is immediate that the radius of B is no less than
√
2/
√
Lemma 4 Let y and z be points on the circle Sx (2/ 3). If each of y and z is equidistant
√
from Ci and C j , then the angle y x z is smaller than or equal to 2 arccos( 3 − 1) =
85.8828 . . .◦ .
Proof Following [
1, 4
], the existence of a supporting hyperplane of Ci that separates
int(Ci ) from int(C j ) suffices.
In [
1
], it is shown that planar objects satisfying Lemmas 2, 3 and 4 have area no less
tπh/a√n1√21i2n. ARsn+th2e. Tbohuenpdrohdouldcst ofofrthalel dDeinriscehldeitskslipcaecsk,iintgfoilnlotwhes tphlaatnδe+w(Dith2 ×RnRgni)ve≤s
a polycylinder packing in Rn+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.
Acknowledgments Thanks to Thomas Hales, Włodzimierz Kuperberg and Robert Kusner. The author
was supported by Austrian Science Fund (FWF) Project 5503 and National Science Foundation (NSF)
Grant No. 1104102.
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