On nicely smooth Banach spaces (Q2732308)

The authors study several properties of smoothness and of intersection of balls in Banach spaces, namely: NEWLINENEWLINENEWLINE1) \(X\) is nicely smooth if \(X^\ast\) contains no proper norming subspace; NEWLINENEWLINENEWLINE2) \(X\) has the ball generated property (BGP) if every closed bounded convex set in \(X\) is the intersection of finite union of (closed) balls; NEWLINENEWLINENEWLINE3) \(X\) has Property (II) if every closed bounded convex set in \(X\) is the intersection of closed convex hulls of finite union of balls (it is equivalent to say, by a result of \textit{D. Chen} and \textit{Bor-Luh Lin} [Rocky Mt. J. Math. 28, No. 3, 835-873 (1998; Zbl 0932.46008)], that the weak-star points of continuity of the unit-ball \(B_{X^\ast}\) of \(X^\ast\) are norm dense in the unit-sphere \(S_{X^\ast}\)); NEWLINENEWLINENEWLINE4) \(X\) has the Mazur intersection property (MIP) if every closed bounded convex set in \(X\) is intersection of balls. Clearly, MIP implies the Property (II) and the BGP. NEWLINENEWLINENEWLINEIt is shown that if \(X^\ast\) is the closed linear span of the points of the unit-ball \(B_{X^\ast}\) which are contained, for every \(\varepsilon>0\), in a convex combination \(S\) of \(w^\ast\)-slices, with \(\text{diam} S\leq \varepsilon\), then \(X\) has the BGP (Proposition 2.5; note that in the proof of Theorem 2.3, the reference to \textit{G. Godefroy} and \textit{N. J. Kalton} [Contemp. Math. 85, 195-237 (1989; Zbl 0676.46003)] is not relevant, since the observation made before is elementary); it follows then, from Bourgain's Lemma, that the Property (II) implies the BGP (Corollary 2.6). The authors give also an elementary proof that the BGP for \(X\) implies that \(X\) is nicely smooth (a result which is originally due to G. Godefroy and N. Kalton, in the above quoted paper).NEWLINENEWLINENEWLINEIn the second part of the paper, the authors study the stability of these properties with respect to sums and tensor products, three-spaces properties, spaces of compact operators, spaces of Banach-valued continuous functions and Bochner-Lebesgue spaces; for instance, it is shown that, for \(1<p<\infty\), the following assertions are equivalent: NEWLINENEWLINENEWLINEa) \(L^p([0,1],X)\) has the \(BGP\); NEWLINENEWLINENEWLINEb) \(L^p([0,1],X)\) is nicely smooth; NEWLINENEWLINENEWLINEc) \(X\) is nicely smooth and Asplund (Theorem 3.9).