A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers (Q1865735)

Let \(c_0(X)\) be the space of all null sequences in a Banach space \(X\). It is shown that each compact set in \(c_0(X)\) admits a Chebyshev center if and only if each compact set in \(X\) admits a center. A Banach space in which all compact sets, but not all bounded sets, admit Chebyshev centers is constructed.