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On Steiner's network problem

Published online by Cambridge University Press:  26 February 2010

J. M. Hammersley
J. M. Hammersley
Affiliation:
Institute of Statistics, Oxford
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Extract

Let S be a point set of the Euclidean plane, such that

(i) S is bounded,

(ii) the closure of S has unit Lebesgue measure.

Let P be an arbitrary set of n points contained in S, and let l(P) denote the total length of the shortest system of lines connecting the points of P together. Define ln to be the supremum of l(P), taken over all sets P of n points in S. Beardwood, Halton, and Hammersley [1[ proved that there exists an absolute constant α, independent of S, such that

Type
Research Article
Information
Mathematika , Volume 8 , Issue 2 , December 1961 , pp. 131 - 132
Copyright
Copyright © University College London 1961

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References

1. Beardwood, Jillian, Halton, J. H., and Hammersley, J. M., “The shortest path through many points”, Proc. Camb. Phil. Soc., 55 (1959), 299–327.CrossRefGoogle Scholar
2. Courant, R. and Robbins, H., What is mathematics? (Oxford University Press, 1941).Google Scholar
3. Few, L., “The shortest path and the shortest road through n points”, Mathematika, 2 (1955), 141–144.CrossRefGoogle Scholar
4. Mirsky, L., “Problems of arithmetical geometry”, Math. Gaz., 44 (1960), 183–191.CrossRefGoogle Scholar
5. Verblunsky, S., “On the shortest path through a number of points”, Proc. American Math. Soc. (2), 6 (1951), 904–913.Google Scholar