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A Characterization of Starshaped Sets

Published online by Cambridge University Press:  20 November 2018

Jean Chan Stanek
Jean Chan Stanek*
Affiliation:
Sonoma State College, Rohnert Park, California 94928
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In 1946 Krasnoselskii proved that if every n + 1 points of a compact connected set S in a Euclidean space En can see at least one point of S via S then S is starshaped [1]. This result was expanded by Robkin in 1965 [2]. In this paper, we characterize starshaped sets in En by a local arcwise convexity property relative to a point p.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 29 , Issue 4 , 01 August 1977 , pp. 673 - 680
Copyright
Copyright © Canadian Mathematical Society 1977

References

1. Krasnoselskii, M. A., Sur un critère pour qu'un domain soit étoile, Mat. Sb., N.S. 19, no. 61, (1946), 309–310.Google Scholar
2. Robkin, E. E., Characterizations of starshaped sets, Doctoral Dissertation, University of California, Los Angeles, 1965.Google Scholar
3. Valentine, F. A., Arcwise convex sets, Proc. Am. Math. Soc. 2 (1951), 159–165.Google Scholar
4. Tietze, H., Uber Konvexheit im kleinen und im grossen und ùber gewisse den Punkten einer Menge zugeordnete Dimensionszahlen, Math. Z. 28 (1928), 697–707.Google Scholar
5. Valentine, F. A., Convex sets (McGraw-Hill Book Company, New York, 1964).Google Scholar