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Saturated systems of symmetric convex domains; results of Eggleston, Bambah and Woods

Published online by Cambridge University Press:  24 October 2008

Vishwa Chander Dumir  and
Dharam Singh Khassa
Vishwa Chander Dumir
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-14.
Dharam Singh Khassa
Affiliation:
Centre for Advanced Study in Mathematics, Panjab University, Chandigarh-14.
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Let K be a closed, bounded, symmetric convex domain with centre at the origin O and gauge function F(x). By a homothetic translate of K with centre a and radius r we mean the set {x: F(x−a) ≤ r}. A family ℳ of homothetic translates of K is called a saturated family or a saturated system if (i) the infimum r of the radii of sets in ℳ is positive and (ii) every homothetic translate of K of radius r intersects some member of ℳ. For a saturated family ℳ of homothetic translates of K, let S denote the point-set union of the interiors of members of ℳ and S(l), the set S ∪ {x: F(x) ≤ l}. The lower density ρℳ(K) of the saturated system ℳ is defined by

where V(S(l)) denotes the Lebesgue measure of the set S(l). The problem is to find the greatest lower bound ρK of ρℳ(K) over all saturated systems ℳ of homothetic translates of K. In case K is a circle, Fejes Tóth(9) conjectured that

where ϑ(K) denotes the density of the thinnest coverings of the plane by translates of K. In part I, we state results already known in this direction. In part II, we prove that ρK = (¼) ϑ(K) when K is strictly convex and in part III, we prove that ρK = (¼) ϑ(K) for all symmetric convex domains.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 74 , Issue 1 , July 1973 , pp. 107 - 116
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

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