For a dense set of equivalent norms, a non-reflexive Banach space contains a triangle with no Chebyshev center

Libor Veselý

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 1, page 153-158
  • ISSN: 0010-2628

Abstract

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Let X be a non-reflexive real Banach space. Then for each norm | · | from a dense set of equivalent norms on X (in the metric of uniform convergence on the unit ball of X ), there exists a three-point set that has no Chebyshev center in ( X , | · | ) . This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.

How to cite

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Veselý, Libor. "For a dense set of equivalent norms, a non-reflexive Banach space contains a triangle with no Chebyshev center." Commentationes Mathematicae Universitatis Carolinae 42.1 (2001): 153-158. <http://eudml.org/doc/248805>.

@article{Veselý2001,
abstract = {Let $X$ be a non-reflexive real Banach space. Then for each norm $|\cdot |$ from a dense set of equivalent norms on $X$ (in the metric of uniform convergence on the unit ball of $X$), there exists a three-point set that has no Chebyshev center in $(X,|\cdot |)$. This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.},
author = {Veselý, Libor},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {renormings; non-reflexive Banach spaces; Chebyshev centers; Chebyshev center; nonreflexive Banach space; renorming; uniform approximation},
language = {eng},
number = {1},
pages = {153-158},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {For a dense set of equivalent norms, a non-reflexive Banach space contains a triangle with no Chebyshev center},
url = {http://eudml.org/doc/248805},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Veselý, Libor
TI - For a dense set of equivalent norms, a non-reflexive Banach space contains a triangle with no Chebyshev center
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 1
SP - 153
EP - 158
AB - Let $X$ be a non-reflexive real Banach space. Then for each norm $|\cdot |$ from a dense set of equivalent norms on $X$ (in the metric of uniform convergence on the unit ball of $X$), there exists a three-point set that has no Chebyshev center in $(X,|\cdot |)$. This result strengthens theorems by Davis and Johnson, van Dulst and Singer, and Konyagin.
LA - eng
KW - renormings; non-reflexive Banach spaces; Chebyshev centers; Chebyshev center; nonreflexive Banach space; renorming; uniform approximation
UR - http://eudml.org/doc/248805
ER -

References

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  1. Davis W.J., Johnson W.B., A renorming of non-reflexive Banach spaces, Israel J. Math. 14 (1973), 353-367. (1973) MR0322481
  2. van Dulst D., Singer I., On Kadec-Klee norms on Banach spaces, Studia Math. 54 (1976), 205-211. (1976) Zbl0321.46012MR0394132
  3. Holmes R.B., A course in optimization and best approximation, Lecture Notes in Mathematics 257, Springer-Verlag, 1972. MR0420367
  4. James R.C., Reflexivity and the supremum of linear functionals, Ann. Math. 66 (1957), 159-169. (1957) Zbl0079.12704MR0090019
  5. Konyagin S.V., A remark on renormings of nonreflexive spaces and the existence of a Chebyshev center, Moscow Univ. Math. Bull. 43 2 (1988), 55-56. (1988) MR0938075
  6. Veselý L., A geometric proof of a theorem about non-dual renormings, Proc. Amer. Math. Soc. 127 (1999), 2807-2809. (1999) MR1670431

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