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.
@article{119230,
author = {Libor Vesel\'y},
title = {For a dense set of equivalent norms, a non-reflexive Banach space contains a triangle with no Chebyshev center},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {153-158},
zbl = {1056.46009},
mrnumber = {1825379},
language = {en},
url = {http://dml.mathdoc.fr/item/119230}
}
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, Tome 42 (2001) pp. 153-158. http://gdmltest.u-ga.fr/item/119230/
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