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1996 | 120 | 3 | 191-204
Tytuł artykułu

Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

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Abstrakty
EN
For combining two convex bodies C and D to produce a third body, two of the most important ways are the operation ∓ of forming the closure of the vector sum C+D and the operation γ̅ of forming the closure of the convex hull of C ⋃ D. When the containing normed linear space X is reflexive, it follows from weak compactness that the vector sum and the convex hull are already closed, and from this it follows that the class of all rotund bodies in X is stable with respect to the operation ∓ and the class of all smooth bodies in X is stable with respect to both ∓ and γ̅. In our paper it is shown that when X is separable, these stability properties of rotundity (resp. smoothness) are actually equivalent to the reflexivity of X. The characterizations remain valid for each nonseparable X that contains a rotund (resp. smooth) body.
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autor
  • Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350, U.S.A., klee@math.washington.edu
Bibliografia
  • [Da] M. M. Day, Strict convexity and smoothness, Trans. Amer. Math. Soc. 78 (1955), 516-528.
  • [Di] J. Diestel, Geometry of Banach Spaces--Selected Topics, Lecture Notes in Math. 485, Springer, Berlin, 1975.
  • [DS] N. Dunford and J. Schwartz, Linear Operators I, Interscience, New York, 1958.
  • [GKM] P. Georgiev, D. Kutzarova and A. Maaden, On the smooth drop property, Nonlinear Anal. 26 (1996), 595-602.
  • [Gr] B. Grünbaum, Convex Polytopes, Wiley-Interscience, London, 1967.
  • [Ja] R. C. James, Reflexivity and the supremum of linear functionals, Ann. of Math. 66 (1957), 159-169.
  • [Kl1] V. Klee, Some new results on smoothness and rotundity in normed linear spaces, Math. Ann. 139 (1959), 51-63.
  • [Kl2] V. Klee, Adjoints of projective transformations and face-figures of convex polytopes, in: Math. Programming Stud. 8, North-Holland, Amsterdam, 1978, 208-216.
  • [Lo] A. R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225-238.
  • [MS] P. McMullen and G. C. Shephard, Convex Polytopes and the Upper Bound Conjecture, London Math. Soc. Lecture Note Ser. 3, Cambridge Univ. Press, 1971.
  • [Ph] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin, 1989.
  • [Ta] M. Talagrand, Renormages de quelques C(K), Israel J. Math. 54 (1986), 327-334.
  • [Tr] S. L. Troyanski, Example of a smooth space whose dual is not strictly convex, Studia Math. 35 (1970), 305-309.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv120i3p191bwm
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