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On the realization of distances within coverings of an n-sphere

Published online by Cambridge University Press:  26 February 2010

D. G. Larman
D. G. Larman
Affiliation:
University College, London.
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Extract

In 1933, K. Borsuk [1] established the well-known result that if n closed sets cover Sn−1 then at least one set contains antipodal points, where Sn−1 is the surface of the ball Tn of centre O and unit diameter in Rn. This result prompted H. Hadwiger [2] to make a still unresolved conjecture which, in the spirit of B. Griinbaum's survey [3], we state as follows: Let r be the largest integer such that whenever r closed sets cover Sn−1 at least one set realizes all distances between 0 and 1. Then r = n.

Type
Research Article
Information
Mathematika , Volume 14 , Issue 2 , December 1967 , pp. 203 - 206
Copyright
Copyright © University College London 1967

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References

1.Borsuk, K., “Drei Sätze über die n-dimensionale Euklidische Sphäre”, Fund. Math., 20 (1933), 177–190.CrossRefGoogle Scholar
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