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A minimal density plane covering problem

Published online by Cambridge University Press:  26 February 2010

H. G. Eggleston
H. G. Eggleston
Affiliation:
Bedford College, Regent's Park, London, N.W.1.
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Extract

We say that a set of closed circular discs of radii r1r2, …, all lying in a Euclidean plane, is saturated if and only if r = inf ri > 0 and any circle of radius r has at least one point in common with a circle of the set. For any set X we use α(X) to denote the area of X. If X denotes the point set union of the discs and X(k) the part of X inside the disc whose centre is the origin and radius k then by the lower density of the covering we mean . The problem is to find the exact lower bound of the lower density for any saturated set of circles. We show that it is φ/(6√3) provided the circles are disjoint. The general case, when they may overlap, remains unsolved.

Type
Research Article
Information
Mathematika , Volume 12 , Issue 2 , December 1965 , pp. 226 - 234
Copyright
Copyright © University College London 1965

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References

1.Tóth, L. Fejes, Lagerungen in der Ebene, auf der Kugel und in Raum, Grundl. Math. Wiss. (Springer, Berlin, 1953).CrossRefGoogle Scholar