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On The Kakeya Constant

Published online by Cambridge University Press:  20 November 2018

F. Cunningham Jr.  and
I. J. Schoenberg
F. Cunningham Jr.
Affiliation:
Bryn Mawr College, The University of Pennsylvania, and Mathematics Research Center, U.S. Army, University of Wisconsin
I. J. Schoenberg
Affiliation:
Bryn Mawr College, The University of Pennsylvania, and Mathematics Research Center, U.S. Army, University of Wisconsin
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We shall say that a plane set D has the Kakeya property if a unit segment can be turned continuously in D through 360° back to its original position. The famous solution of this problem by A. S. Besicovitch (1; 2; 4; 5; 6), to the effect that there are sets of arbitrarily small area having the Kayeka property, leaves open the problem obtained by adding the new condition that the set D be also simply connected. Since we do not know whether there is an attainable minimum, we define the Kakeya constant K to be the greatest lower bound of areas of simply connected sets having the Kakeya property. We shall refer to such sets as Kakeya sets.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 946 - 956
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Besicovitch, A. S., On Kakeyas problem and a similar one, Math. Z., 27 (1928), 312–320.Google Scholar
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7. Walker, R. J., Addendum to Mr. Greenwood's paper, Pi Mu Epsilon J. (November 1952), 275Google Scholar