Quasi uniform convexity : Revisited

Quasi uniform convexity (QUC) is a geometric property of Banach spaces, introduced in 1973 by J.R. Calder et al., which implies existence of Chebyshev centers for bounded sets. We extend and strengthen some known results about this property. We show that (QUC) is equivalent to existence and continuous dependence (in the Hausdorff metric) of Chebyshev centers of bounded sets. If X is (QUC) then the space C(K;X) of continuous X-valued functions on a compact K is (QUC) as well. We also show that a sufficient condition introduced by L. Pevac already implies (QUC), and we provide a couple of new sufficient conditions for (QUC). Together with Chebyshev centers, we consider also asymptotic centers for bounded sequences or nets (of points or sets).