  In this paper we address the question whether in a given Banach space, a
Chebyshev center of a nonempty bounded subset can be a farthest point of the
set. Our exploration reveals that the answer depends on the convexity
properties of the Banach space. We obtain a characterization of two-dimensional
real strictly convex spaces in terms of Chebyshev center not contributing to
the set of farthest points. We explore the scenario in uniformly convex Banach
spaces and further study the roles played by centerability and M-compactness in
the scheme of things to obtain a step by step characterization of strictly
convex Banach spaces. We also illustrate with examples the optimality of our
results.
