# Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

Victor Klee; Libor Veselý; Clemente Zanco

Studia Mathematica (1996)

- Volume: 120, Issue: 3, page 191-204
- ISSN: 0039-3223

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topKlee, Victor, Veselý, Libor, and Zanco, Clemente. "Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces." Studia Mathematica 120.3 (1996): 191-204. <http://eudml.org/doc/216331>.

@article{Klee1996,

abstract = {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.},

author = {Klee, Victor, Veselý, Libor, Zanco, Clemente},

journal = {Studia Mathematica},

keywords = {normed linear space; reflexive; convex body; smooth; rotund; strictly convex; vector sum; convex hull; stability; convex bodies; closure of the vector sum; weak compactness; stability properties of rotundity; smoothness},

language = {eng},

number = {3},

pages = {191-204},

title = {Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces},

url = {http://eudml.org/doc/216331},

volume = {120},

year = {1996},

}

TY - JOUR

AU - Klee, Victor

AU - Veselý, Libor

AU - Zanco, Clemente

TI - Rotundity and smoothness of convex bodies in reflexive and nonreflexive spaces

JO - Studia Mathematica

PY - 1996

VL - 120

IS - 3

SP - 191

EP - 204

AB - 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.

LA - eng

KW - normed linear space; reflexive; convex body; smooth; rotund; strictly convex; vector sum; convex hull; stability; convex bodies; closure of the vector sum; weak compactness; stability properties of rotundity; smoothness

UR - http://eudml.org/doc/216331

ER -

## References

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