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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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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.
  • Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195-4350, U.S.A.
  • Dipartimento di Matematica, Università degli Studi, via C. Saldini 50, 20133 Milano MI, Italy
  • Dipartimento di Matematica, Università degli Studi, via C. Saldini 50, 20133 Milano MI, Italy
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