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A theorem on convex kernels

Published online by Cambridge University Press:  26 February 2010

Victor Klee
Victor Klee
Affiliation:
University of Washington, Seattle, Washington, U.S.A.
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For a subset S of a real linear space, let ck S denote the set of all points from which S is starshaped; that is p∈ ck S if and only if S contains the segment [p, s] for all s∈ S. The set ck S, which is necessarily convex, was introduced by H. Brunn [2[ in 1913 as the Kerneigebeit or convex kernel of the set S. Of course ck S = S if and only if the set S itself is convex. L. Fejes Tóth asked for a characterization of those plane convex bodies which can be realized as the convex kernels of nonconvex plane domains, and it was proved by N. G. de Bruijn and K. Post that every plane convex body can be so realized. Here we establish a stronger result.

Type
Research Article
Information
Mathematika , Volume 12 , Issue 1 , June 1965 , pp. 89 - 93
Copyright
Copyright © University College London 1965

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References

1. Bishop, E. and Phelps, K., “The support functionals of a convex set”, Proceedings of Symposia in Pure Math. (American Math. Soc), 7 (1963), 27–35.Google Scholar
2. Brunn, H., “Über Kerneigebiete”, Math. Annalen, 73 (1913), 436–440.CrossRefGoogle Scholar
3. Klee, V., “Extremal structure of convex sets. II”, Math. Zeitschrift, 69 (1958), 90–104.CrossRefGoogle Scholar
4. Post, K., “Star extension of plane convex sets”, Indag. Math., 26 (1964), 330–338.CrossRefGoogle Scholar