Packing density

A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. In packing problems, the objective is usually to obtain a packing of the greatest possible density.

In compact spaces

If

K

1

,

…

,

K

n

{\displaystyle K_{1},\dots ,K_{n}}

are measurable subsets of a compact measure space

X

{\displaystyle X}

and their interiors pairwise do not intersect, then the collection

[

K

i

]

{\displaystyle [K_{i}]}

is a packing in

X

{\displaystyle X}

and its packing density is

η

=

∑

i

=

1

n

μ

(

K

i

)

μ

(

X

)

.

{\displaystyle \eta ={\frac {\sum _{i=1}^{n}\mu (K_{i})}{\mu (X)}}.}

In Euclidean space

If the space being packed is infinite in measure, such as Euclidean space, it is customary to define the density as the limit of densities exhibited in balls of larger and larger radii. If

B

t

{\displaystyle B_{t}}

is the ball of radius

t

{\displaystyle t}

centered at the origin, then the density of a packing

[

K

i

:

i

∈

N

]

{\displaystyle [K_{i}:i\in \mathbb {N} ]}

is

η

=

lim

t

→

∞

∑

i

=

1

∞

μ

(

K

i

∩

B

t

)

μ

(

B

t

)

.

{\displaystyle \eta =\lim _{t\to \infty }{\frac {\sum _{i=1}^{\infty }\mu (K_{i}\cap B_{t})}{\mu (B_{t})}}.}

Since this limit does not always exist, it is also useful to define the upper and lower densities as the limit superior and limit inferior of the above respectively. If the density exists, the upper and lower densities are equal. Provided that any ball of the Euclidean space intersects only finitely many elements of the packing and that the diameters of the elements are bounded from above, the (upper, lower) density does not depend on the choice of origin, and

μ

(

K

i

∩

B

t

)

{\displaystyle \mu (K_{i}\cap B_{t})}

can be replaced by

μ

(

K

i

)

{\displaystyle \mu (K_{i})}

for every element that intersects

B

t

{\displaystyle B_{t}}

.

The ball may also be replaced by dilations of some other convex body, but in general the resulting densities are not equal.