Given a Banach spaceX, letc 0(X) be the space of all null sequences inX (equipped with the supremum norm). We show that: 1) each compact set inc 0(X) admits a (Chebyshev) center iff each compact set inX admits a center; 2) forX satisfying a certain condition (Q), each bounded set inc 0(X) admits a center iffX is quasi uniformly rotund. We construct a Banach spaceX such that the compact subsets ofX admit centers,X satisfies the condition (Q) andX is not quasi uniformly rotund. It follows that the Banach spaceE=c 0(X) has the property from the title.