In this paper some properties of a special type of boundary point of convex sets in Banach spaces are studied. Specifically, a strongly extreme point x of a convex set S is a point of S such that for each real number r>0, segments of length 2r and centered x are not uniformly closer to S than some positive number d(x,r). Results are obtained comparing the notion of strongly extreme point to other known types of special boundary points of convex sets. Using the notion of strongly extreme point, a convexity condition is defined on the norm of the space under consideration, and this convexity condition makes possible a unified treatment of some previously studied convexity conditions. In addition, a sufficient condition is given on the norm of a separable conjugate space for every extreme point of the unit ball to be strongly extreme.
KeywordsBanach Space Real Number Number Theory Extreme Point Boundary Point
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- 1.DAY, M. M.: Some characterizations of inner product spaces, Trans. Amer. Math. Soc. 62, 320–337 (1947).Google Scholar
- 2.—: Normed Linear Spaces, Academic Press, New York, Second Printing, 1962.Google Scholar
- 3.DUNFORD, N. and SCHWARTZ, J. T.,: Linear Operators, Part I, Interscience, New York, Second Printing, 1964.Google Scholar
- 4.JAMES, R. C.: Uniformly non-square Banach spaces, Annals of Math. 80, 542–550 (1964).Google Scholar
- 5.LINDENSTRAUSS, J.: On operators which attain their norm, Israel J. Math. 1, 139–148 (1963).Google Scholar
- 6.MILMAN, V. D.: Infinite-dimensional geometry of the unit sphere in Banach space, Soviet Math. Dokl. 8, 1440–1444 (1967).Google Scholar
- 7.NAMIOKA, I.: Neighborhoods of extreme points, Israel J. Math. 5, 145–152 (1967).Google Scholar
- 8.RIEFFEL, M. S.: Dentable subsets of Banach spaces, with applications to a Radon-Nikodym theorem, in Functional Analysis, proceedings of a conference held at the U. of Calif. at Irvine, edited by B. Gelbaum, Thompson Book Company, Washington, D. C., 1967.Google Scholar