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A conjecture of Fejes Tóth on saturated systems of circles

Published online by Cambridge University Press:  24 October 2008

Vishwa Chander Dumir  and
Dharam Singh Khassa
Vishwa Chander Dumir
Affiliation:
Department of Mathematics, Punjab University, Chandigarh 14, India
Dharam Singh Khassa
Affiliation:
Department of Mathematics, Punjab University, Chandigarh 14, India
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Extract

A family ℳ of closed circular discs in the plane is called a saturated family or a saturated system of circles if (i) the infimum r of the radii of the discs in ℳ is positive and (ii) every closed disc of radius r in the plane intersects at least one disc in ℳ. For a saturated family ℳ, we denote by S the point-set union of the interiors of the members of ℳ and by S(l) the part of S inside the circular disc of radius l centred at the origin. We define the lower density ρℳ of the saturated family ℳ as

where V(S(l)) denotes the Lebesgue measure of the set S(l).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 3 , November 1973 , pp. 453 - 460
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

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