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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
2.Hadwiger, H., “Ein Überdeckungssatz für den Euklidischen Raum”, Portugal. Math., 4 (1944), 140–144.Google Scholar
3.Grunbaum, B., “Borsuk's problem and related questions”, Proc. Symp. in Pure Math. VII. Convexity (American Math. Soc, 1963).Google Scholar
4.Hadwiger, H., “Eine Bermerkung zum Borsukschen Antipodensatz”, Vierteljschr. Naturf. Ges. Zürich, 89 (1944), 211–214.Google Scholar
5.Hopf, H., “Eine Verallgemeinerung bekannter Abbildungs und Uberdeckungssütze”, Portugal-Math., 4 (1944), 129–139.Google Scholar
6.Lyusternik, L. A. and Schnirelman, L. G., “Topological methods in variational problems and their application to the differential geometry of surfaces”, Usp. Mat. Nauk (N.S.), 2, No. 1, 17(1947), 166–217.Google Scholar