For a dense set of equivalent norms, a non-reflexive Banach space contains a triangle with no Chebyshev center
Libor Veselý · Czech digital mathematics library · 2001
summary: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.