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A Property of Plane Sets of Constant Width

Published online by Cambridge University Press:  20 November 2018

Z.A. Melzak
Z.A. Melzak*
Affiliation:
McGill University
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It is well known that sets of constant width share several properties with spheres. In this note we consider a simple property of the circle and we show that it is shared by every plane set of constant width. As an application we derive a stronger form of the following theorem of D. Gale, [1]: every plane set of diameter 1 is a union of three sets of diameters not exceeding √3/2, and this constant is best possible. We shall make free use of the more elementary properties of convex sets and of sets of constant width; for these properties and for the terminology see the standard reference [2], or [3].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 6 , Issue 3 , 01 September 1963 , pp. 409 - 415
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Gale, D., Proc. A. M. S., 4(1953), 222-225.Google Scholar
2. Bonnesen, T. and Fenchel, W., Theorie der konvexen Koerper, Leipzig, (1934).Google Scholar
3. Eggleston, H., Convexity, No. 47, Cambridge Tracts, (1958).Google Scholar
4. Borsuk, K., Fund. Math., 20(1933), 177-190.Google Scholar
5. Gruenbaum, B., Proc. Cambr. Phil. Soc., 53(1957), 776-778.Google Scholar