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A Packing Inequality for Compact Convex Subsets of the Plane

Published online by Cambridge University Press:  20 November 2018

J. H. Folkman  and
R. L. Graham
J. H. Folkman
Affiliation:
The Rand Corporation, Santa MonicaCalifornia
R. L. Graham
Affiliation:
Bell Telephone Laboratories, Incorporated Murray HillNew Jersey
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Let X be a compact metric space. By a packing in X we mean a subset S ⊆ X such that, for x, y ∈ S with x ≠ y, the distance d(x, y) ≥ 1. Since X is compact, any packing of X is finite. In fact, the set of numbers

{card(S): S is a packing in X}

is bounded. The cardinality of the largest packing in X will be called the packing number of X and will be denoted by ρ(X). If A(X) and P(X) denote the area and perimeter, respectively, of a compact convex subset X of the plane, then a special case of a result conjectured by H. Zassenhaus [6] and proved by N. Oler [l] is the following.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 6 , December 1969 , pp. 745 - 752
Copyright
Copyright © Canadian Mathematical Society 1969

References

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5. Pontryagin, L.S., Combinatorial topology. (Graylock, New York, 1952)Google Scholar
6. Zassenhaus, H., Modern development in the geometry of numbers. Bull. Amer. Math. Soc. 67 (1961) 427–439.Google Scholar