Geodesic
Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces
Studia Mathematica, Tome 120 (1996) no. 3, pp. 191-204

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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.
DOI : 10.4064/sm-120-3-191-204
Keywords: normed linear space, reflexive, convex body, smooth, rotund, strictly convex, vector sum, convex hull, stability

Victor Klee 1 ; L. Veselý 1 ; C. Zanco 1

1 
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Victor Klee; L. Veselý; C. Zanco. Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces. Studia Mathematica, Tome 120 (1996) no. 3, pp. 191-204. doi: 10.4064/sm-120-3-191-204

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