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.
@article{bwmeta1.element.bwnjournal-article-smv120i3p191bwm,
author = {Victor Klee and Libor Vesel\'y and Clemente Zanco},
title = {Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces},
journal = {Studia Mathematica},
volume = {119},
year = {1996},
pages = {191-204},
zbl = {0863.46004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv120i3p191bwm}
}
Klee, Victor; Veselý, Libor; Zanco, Clemente. Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces. Studia Mathematica, Tome 119 (1996) pp. 191-204. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv120i3p191bwm/
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