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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
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of the above respectively. If the density exists, the upper and lower densities are equal. Provided that any ball of the
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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. The
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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
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of a fixed body is also a common supply collection of interest, and it defines the translative packing constant of that body.
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347:{\displaystyle \eta =\lim _{t\to \infty }{\frac {\sum _{i=1}^{\infty }\mu (K_{i}\cap B_{t})}{\mu (B_{t})}}}
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Since this limit does not always exist, it is also useful to define the upper and lower densities as the
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and their interiors pairwise do not intersect, then the collection is a packing in
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Groemer, H. (1986), "Some basic properties of packing and covering constants",
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states that 3-balls have the lowest packing constant of any convex solid. All
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156:{\displaystyle \eta ={\frac {\sum _{i=1}^{n}\mu (K_{i})}{\mu (X)}}}
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455:. The
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