Geodesic
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) no. 1, pp. 153-158.

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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.
Classification : 41A65, 46B03, 46B20
Keywords: renormings; non-reflexive Banach spaces; Chebyshev centers 
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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, Tome 42 (2001) no. 1, pp. 153-158. http://geodesic.mathdoc.fr/item/CMUC_2001__42_1_a10/