Abstract
Given a Banach spaceX, letc0(X) be the space of all null sequences inX (equipped with the supremum norm). We show that: 1) each compact set inc0(X) admits a (Chebyshev) center iff each compact set inX admits a center; 2) forX satisfying a certain condition (Q), each bounded set inc0(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=c0(X) has the property from the title.
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