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Apacking density orpacking fraction of a packing in some space is thefraction 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. Inpacking problems, the objective is usually to obtain a packing of the greatest possible density.

If${\displaystyle K_1,\dots ,K_n}$ are measurable subsets of acompactmeasure space${\displaystyle X}$and their interiors pairwise do not intersect, then the collection${\displaystyle [K_i]}$ is a packing in${\displaystyle X}$ and its packing density is$${\displaystyle \eta =\frac{\sum _{i=1}^n\mu (K_i)}{\mu (X)}.}$$

If the space being packed is infinite in measure, such asEuclidean space, it is customary to define the density as the limit of densities exhibited in balls of larger and larger radii. If${\displaystyle B_t}$ is the ball of radius${\displaystyle t}$ centered at the origin, then the density of a packing${\displaystyle [K_i:i\in \mathbb{N}]}$ is$${\displaystyle \eta =\underset{t\to \mathrm{\infty }}{lim}\frac{\sum _{i=1}^{\mathrm{\infty }}\mu (K_i\cap B_t)}{\mu (B_t)}.}$$Since this limit does not always exist, it is also useful to define the upper and lower densities as thelimit 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${\displaystyle \mu (K_i\cap B_t)}$ can be replaced by${\displaystyle \mu (K_i)}$ for every element that intersects${\displaystyle B_t}$.^[1]The ball may also be replaced by dilations of some other convex body, but in general the resulting densities are not equal.

One is often interested in packings restricted to use elements of a certain supply collection. For example, the supply collection may be the set of all balls of a given radius. Theoptimal packing density orpacking constant associated with a supply collection is thesupremum of upper densities obtained by packings that are subcollections of the supply collection. If the supply collection consists of convex bodies of bounded diameter, there exists a packing whose packing density is equal to the packing constant, and this packing constant does not vary if the balls in the definition of density are replaced by dilations of some other convex body.^[1]

A particular supply collection of interest is allEuclidean motions of a fixed convex body${\displaystyle K}$. In this case, we call the packing constant the packing constant of${\displaystyle K}$. TheKepler conjecture is concerned with the packing constant of 3-balls.Ulam's packing conjecture states that 3-balls have the lowest packing constant of any convex solid. Alltranslations of a fixed body is also a common supply collection of interest, and it defines the translative packing constant of that body.

• Atomic packing factor
• Sphere packing
• List of shapes with known packing constant

• Weisstein, Eric W."Packing Density".MathWorld.

• Packing problems
• Discrete geometry

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