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

Abstract

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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.

How to cite

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Klee, 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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  9. [Lo] A. R. Lovaglia, Locally uniformly convex Banach spaces, Trans. Amer. Math. Soc. 78 (1955), 225-238. Zbl0064.35601
  10. [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. Zbl0217.46702
  11. [Ph] R. R. Phelps, Convex Functions, Monotone Operators and Differentiability, Lecture Notes in Math. 1364, Springer, Berlin, 1989. Zbl0658.46035
  12. [Ta] M. Talagrand, Renormages de quelques C(K), Israel J. Math. 54 (1986), 327-334. Zbl0611.46023
  13. [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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