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The convex generation of convex Borel sets in Banach spaces

Part of: General convexity

Published online by Cambridge University Press:  26 February 2010

D. Preiss
D. Preiss
Affiliation:
Charles University, Prague, Czechoslovakia.
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Extract

In this note we prove that every convex Borel set in a finite dimensional real Banach space can be obtained, starting from the compact convex sets, by the iteration of countable increasing unions and countable decreasing intersections. This question was first raised by V. Klee [1, p. 451]. It was answered affirmatively by Klee for R2 in [2, pp. 109–111] and for R3 by D. G. Larman in [4]. C. A. Rogers has given an equivalent formulation of the question for Rn in [6].

MSC classification

Secondary: 52A20: Convex sets in $n$ dimensions (including convex hypersurfaces)
Type
Research Article
Information
Mathematika , Volume 20 , Issue 1 , June 1973 , pp. 1 - 3
Copyright
Copyright © University College London 1973

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References

1. Klee, V., “Convex sets in linear spaces”, Duke Math. J., 18 (1951), 443–466.Google Scholar
2. Klee, V., “Convex sets in linear spaces, III”, Duke Math. J., 20 (1953), 875–883.Google Scholar
3. Kuratowski, K., Topology, Vol. 1 (Academic Press, New York, 1966).Google Scholar
4. Larman, D. G., “The convex Borel sets in R3 are convexly generated”, J. London Math. Soc., 2 (1971), 5–14.Google Scholar
5. Rockafellar, R. T., Convex Analysis (Princeton Univ. Press, Princeton, 1970).Google Scholar
6. Rogers, C. A., “The convex generation of convex Borel sets in Euclidean space”, Pacific J. Math., 35 (1970), 773–782.Google Scholar
7. Yosida, K., Functional Analysis, 3rd Edition (Springer-Verlag, Berlin, 1971).Google Scholar