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A Note on the Borsuk Conjecture

Published online by Cambridge University Press:  20 November 2018

Z.A. Melzak
Z.A. Melzak*
Affiliation:
University of British Columbia Courant Institute of Mathematical Sciences
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According to the still unproved conjecture of Borsuk [1] a bounded subset A of the Euclidean n-space En is a union of n + 1 sets of diameters less than the diameter D of A. Since A can be imbedded in a set of constant width D, [2], it may be assumed that A is already of constant width. If in addition A is smooth, i. e., if through every point of its boundary ∂A there passes one and only one support plane of A, then the truth of Borsuk′s conjecture can be proved very easily [3]. The question arises whether Borsuk′s conjecture holds also for arbitrary smooth convex bodies, not merely for those of constant width. Since it is not known whether a smooth convex body K can be imbedded in a smooth set of constant width D, the answer is not immediate. In this note we show that the answer is affirmative.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 10 , Issue 1 , April 1967 , pp. 1 - 3
Copyright
Copyright © Canadian Mathematical Society 1967

Footnotes

1

During the writing of this note the author held a Fellowship of the National Research Council.

References

1. Borsuk, K., Drei Satze ueber die n-dimensionale Euklidi sche Sphaere. Fund. Math. vol. 20 (1933), pages 177-190.CrossRefGoogle Scholar
2. Bonnesen, T. and Fenchel, W., Theorie der Konvexen Koerper. Chelsea, New York (1944).Google Scholar
3. Hadwiger, H., Mitteilung. Comment. Math. Helv. vol. 19 (1946-47), pages 72-73.CrossRefGoogle Scholar