Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On points at which a set is cone-shaped

Authors: M. Edelstein, L. Keener and R. O’Brien
Journal: Proc. Amer. Math. Soc. 66 (1977), 327-330
MSC: Primary 46B05; Secondary 52A05
MathSciNet review: 0454593
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A set $ \mathcal{S}$ in a normed linear space X is said to be cone-shaped at $ x \in X$ if there is a closed half-space that has x in its bounding hyperplane and contains $ \{ y \in \mathcal{S}:[x,y] \subset S\} $. The point x is called a cone point. In this paper it is shown that if X has an equivalent uniformly convex and uniformly smooth norm and if $ \mathcal{S}$ is a closed bounded subset with the finite visibility property for cone points (i.e., for every finite set F of cone points of S there is a point $ z \in S$ such that $ [z,y] \subset \mathcal{S}$ for all $ y \in F$), then S is starshaped.

References [Enhancements On Off] (What's this?)

  • [1] R. Alexander and M. Edelstein, Finite visibility and starshape in Hilbert space (preprint).
  • [2] D. Amir and J. Lindenstrauss, The structure of weakly compact sets in Banach spaces, Ann. of Math. (2) 88 (1968), 35-46. MR 0228983 (37:4562)
  • [3] J. Borwein, M. Edelstein and R. O'Brien, Visibility and starshape, J. London Math. Soc. (2) 14 (1976), 313-318. MR 0428010 (55:1040)
  • [4] M. M. Day, Normed linear spaces, third ed., Springer-Verlag, Berlin and New York, 1973. MR 0344849 (49:9588)
  • [5] J. Diestel, Geometry of Banach spaces-selected topics, Springer-Verlag, Berlin and New York, 1975. MR 0461094 (57:1079)
  • [6] M. Edelstein, On nearest points in uniformly convex Banach spaces, J. London Math. Soc. 43 (1967), 375-377. MR 0226364 (37:1954)
  • [7] M. Edelstein and L. Keener, Characterizations of infinite-dimensional and nonreflexive spaces, Pacific J. Math. 57 (1975), 365-369. MR 0383046 (52:3927)
  • [8] V. Klee, Extremal structure of convex sets. II, Math. Z. 69 (1958), 90-104. MR 0092113 (19:1065b)
  • [9] M. Krasnoselski, Sur un critère pour qu'un domaine soit étoelée, Mat. Sb. 19 (1946), 309-310. MR 0020248 (8:525a)
  • [10] N. T. Peck, Support points in locally convex spaces, Duke Math. J. 38 (1971), 271-278. MR 0282191 (43:7904)
  • [11] S. B. Stečkin, Approximation properties of sets in normed linear spaces, Rev. Math. Pures Appl. 8 (1963), 5-18. (Russian) MR 0155168 (27:5108)
  • [12] F. Valentine, Convex sets, McGraw-Hill, New York, 1964. MR 0170264 (30:503)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 46B05, 52A05

Retrieve articles in all journals with MSC: 46B05, 52A05

Additional Information

Keywords: Starshaped, superreflexive, finite visibility
Article copyright: © Copyright 1977 American Mathematical Society

American Mathematical Society