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The Packing of Spheres in the Space lp

Published online by Cambridge University Press:  18 May 2009

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A point x in the real or complex space lpis an infinite sequence,(x1, x2, x3,…) of real or complex numbers such that is convergent. Here p ≥ 1 and we write

The unit sphere S consists of all points x ε lp for which ¶ x ¶ ≤ 1. The sphere of radius a≥ ≤ 0 and centre y is denoted by Sa(y) and consists of all points x ε lp such that ¶ x - y ¶ ≤ a. The sphere Sa(y) is contained in S if and only if ¶ y ¶≤1 - a, and the two spheres Sa(y) and Sa(z) do not overlap if and only if

¶ y- z ¶≥ 2a

The statement that a finite or infinite number of spheres Sa (y) of fixed radius a can be packed in S means that each sphere Sa (y) is contained in S and that no two such spheres overlap.

Type
Research Article
Copyright
Copyright © Glasgow Mathematical Journal Trust 1958

References

1.Rankin, R. A., On sums of powers of linear forms I, Ann. of Math., 50 (1949), 691698.CrossRefGoogle Scholar
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3.Rankin, R. A., On packings of spheres in Hilbert space, Proc. Glasgow Math. Assoc., 2 (1955), 145146.CrossRefGoogle Scholar