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The realization of distances within sets in Euclidean space

Part of: Designs and configurations

Published online by Cambridge University Press:  26 February 2010

D. G. Larman  and
C. A. Rogers
D. G. Larman
Affiliation:
Department of Mathematics, University College London.
C. A. Rogers
Affiliation:
Department of Mathematics, University College London.
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Extract

In 1944 and 1945 H. Hadwiger [1, 2] proved the following theorems.

Theorem A. Let En be covered by n + 1 closed sets. Then there is one of the sets, within which all distances are realized.

Theorem B. Let En be covered by 4n−3 closed sets that are all mutually congruent. Then all distances are realized within each set.

Here a distance d is realized within a set S, if there are points x, y in S at distance d apart.

MSC classification

Secondary: 05B99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 19 , Issue 1 , June 1972 , pp. 1 - 24
Copyright
Copyright © University College London 1972

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References

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