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.
Keywords: renormings, non-reflexive Banach spaces, Chebyshev centers
AMS Subject Classification: Primary 46B03; Secondary 41A65