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On a Question of Colin Clark Concerning Three Properties of Convex Sets(1)

Published online by Cambridge University Press:  20 November 2018

Victor Klee
Victor Klee*
Affiliation:
University of Washington, Seattle, Washington
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Let C be a convex subset of a normed linear space E. The following properties of C were studied by Clark [2]:

  1. (1) C is of finite width/ that is, C lies between two parallel closed hyperplanes;

  2. (2) C is of finite width in some direction; that is, for some line L in E there is a finite upper bound for the lengths of C's intersections with lines parallel to L;

  3. (3) C is partially bounded; that is, there is a finite upper bound for the radii of balls contained in C.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 4 , December 1972 , pp. 535 - 537
Copyright
Copyright © Canadian Mathematical Society 1972

Footnotes

(1)

Preparation of this paper was supported in part by the Office of Naval Research.

References

1. Bourbaki, N., Espaces vectorielles topologiques, Ch. 1-2, Hermann, Paris, 1966.Google Scholar
2. Clark, C., On convex sets of finite width, J. London Math. Soc. 43 (1968), 513-516.Google Scholar
3. Klee, V., Convex sets in linear spaces, Duke Math. J. 18 (1951), 443-466.Google Scholar
4. Thorp, E. and Whitley, R., Partially bounded sets of infinite width, J. Reine Angew. Math. 248(1971), 117-122.Google Scholar