Smith's counterexample about uniform rotundity in every direction (Q2708475)

A Banach space \(X\) is called uniformly rotund in every direction (URED) if \(\inf \{1-2^{-1}\|x+y\|: x,y \in B_X, x-y = \lambda z, \|x-y\|\geq \epsilon\} > 0, \) for every \( z \in X\), \(z\neq 0.\) URED spaces were considered by \textit{A. L. Garkavi} [Izv. Akad. Nauk. SSSR, Ser. Mat. 26, 87-106 (1962), English translation: Am. Math. Soc. Transl., II. Ser. 39, 111-132 (1964; Zbl 0158.13602)], in order to characterize normed spaces in which every bounded set has at least one Chebyshev center. Let \(E\) be a sequence Banach space which is a full function space, i.e., \(\alpha = (\alpha _i)\), \(\beta = (\beta _i) \in E\) and \(|\alpha _i|\leq |\beta_i|, \) for all \(i \in \mathbb N\), implies \(\|\alpha\|\leq \|\beta\|\) [see \textit{M. M. Day}, ``Normed linear spaces'', Third Edition, Springer Verlag, Berlin (1973; Zbl 0268.46013)]. If \(X_i\), \(i\in \mathbb N,\) are normed spaces and \(E\) is as above, then one defines the space \(E(X_i)\) by \(E(X_i) = \{(x_i) : x_i \in X_i, i \in \mathbb N,\) and \( (\|x_i\|)\in E\}\), with the norm \(\|(x_i)\|= \|(\|x_i\|)\|_E.\) \textit{M. A. Smith} [Pac. J. Math. 73, No. 1, 215-219 (1977; Zbl 0697.54017)], showed that, in general, \(E\) and each \(X_i\) URED do not imply that \(E(X_i)\) is URED. Sufficient conditions for the validity of this implication are: \(E\) be URED and have compact order intervals (M. A. Smith, loc. cit.), or \(E\) be weakly uniformly rotund with respect to its evaluation functionals (M. M. Day, loc. cit.). For a large class of full function spaces \(E\), including those considered by M. A. Smith and M. M. Day, the authors give necessary and sufficient conditions for the validity of the equivalence: \(E\) and all \(X_i\) are URED if and only if \(E(X_i)\) is URED.