A characterization theorem for bounded starshaped sets in the plane
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- by Marilyn Breen PDF
- Proc. Amer. Math. Soc. 94 (1985), 693-698 Request permission
Abstract:
Let $S$ be a nonempty bounded set in ${R^2}$. Then $S$ is starshaped if and only if every 3 or fewer boundary points of $S$ are clearly visible via $S$ from a common point of $S$. The number 3 is best possible.References
- Marilyn Breen, A Krasnosel′skiĭ-type theorem for points of local nonconvexity, Proc. Amer. Math. Soc. 85 (1982), no. 2, 261–266. MR 652454, DOI 10.1090/S0002-9939-1982-0652454-2
- Marilyn Breen, Points of local nonconvexity and sets which are almost starshaped, Geom. Dedicata 13 (1982), no. 2, 201–213. MR 684155, DOI 10.1007/BF00147663
- Marilyn Breen, The dimension of the kernel of a planar set, Pacific J. Math. 82 (1979), no. 1, 15–21. MR 549829
- K. J. Falconer, The dimension of the convex kernel of a compact starshaped set, Bull. London Math. Soc. 9 (1977), no. 3, 313–316. MR 467536, DOI 10.1112/blms/9.3.313
- W. R. Hare Jr. and John W. Kenelly, Intersections of maximal starshaped sets, Proc. Amer. Math. Soc. 19 (1968), 1299–1302. MR 233283, DOI 10.1090/S0002-9939-1968-0233283-3
- M. Krasnosselsky, Sur un critère pour qu’un domaine soit étoilé, Rec. Math. [Mat. Sbornik] N. S. 19(61) (1946), 309–310 (Russian, with French summary). MR 0020248
- Nick M. Stavrakas, The dimension of the convex kernel and points of local nonconvexity, Proc. Amer. Math. Soc. 34 (1972), 222–224. MR 298549, DOI 10.1090/S0002-9939-1972-0298549-0
- Frederick A. Valentine, Convex sets, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Toronto-London, 1964. MR 0170264
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 94 (1985), 693-698
- MSC: Primary 52A30; Secondary 52A35
- DOI: https://doi.org/10.1090/S0002-9939-1985-0792285-9
- MathSciNet review: 792285