We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

We present a construction that enables one to find Banach spaces $X$ whose sets $\operatorname{NA}(X)$ of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently, $X$ does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2, Israel J. Math. (to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces, J. Funct. Anal. 272 (2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space $X$ where the set $\operatorname{NA}(X)$ for the original norm is not “too large”. The construction can be applied to every Banach space containing $c_{0}$ and having a countable system of norming functionals, in particular, to separable Banach spaces containing $c_{0}$ . We also provide some geometric properties of the norms we have constructed.

The aim of this note is to study octahedrality in vector-valued Lipschitz-free Banach spaces on a metric space, under topological hypotheses on it, by analysing the weak-star strong diameter 2 property in Lipschitz function spaces. Also, we show an example that proves that our results are optimal and that octahedrality in vector-valued Lipschitz-free Banach spaces actually relies on the underlying metric space as well as on the Banach one.

We obtain an existence-uniqueness result for a second order Neumann boundary value problem including cases where the nonlinearity possibly crosses several points of resonance. Optimal and Schauder fixed points methods are used to prove this kind of results.

Linseed (LO) and soyabean (SO) oils were evaluated as fish-oil (FO) substitutes in the diets of marketable-sized gilthead sea bream (Sparus aurata). Practical diets were designed factorially with the lipid added as follows (%): FO 100, LO 60+FO 40, LO 80+FO 20, SO 60+FO 40, SO 80+FO 20. The effects of experimental diets on growth, fatty acids patterns in liver and muscle, flesh quality variables and activities of selected enzymes involved in lipid synthesis and catabolism were determined at the end of a 7-month trial. Fatty acid composition of liver and muscle generally reflected the fatty acid composition of the diets. The n−3 PUFA levels were significantly reduced by the inclusion of vegetable oils. This tendency was more pronounced for EPA than for docosahexaenoic acid. The n−3:n−6 fatty acid ratio reached the lowest values in fish fed the SO diets; this was associated with a higher liver lipid deposition. No differences were found in fillet texture and pH. However, under conditions of forced peroxidation, muscles from fish fed the SO diets had lower peroxidation levels. Vegetable oil substitution decreased lipogenesis in liver and this effect was greatest at the highest substitution level. In contrast, muscle β-oxidation enzymes had increased activities with vegetable oil substitution. Thus, the lower hepatic lipogenesis was correlated with an increased lipid utilisation in muscle. It is concluded that growth and lipid metabolism were affected by experimental diets.

Let $A$ be an infinite-dimensional $C^*$-algebra. It is proved that every nonempty relatively weakly open subset of the closed unit ball $B_A$ of $A$ has diameter equal to 2. This implies that $B_A$ is not dentable, and that there is not any point of continuity for the identity mapping $(B_A,{\rm weak)\,{\longrightarrow}\,(B_A,{\rm norm})$.

We show that an infinite-dimensional real Banach space with numerical index 1 satisfying the Radon–Nikodym property contains l1. It follows that a reflexive or quasi-reflexive real Banach space cannot be re-normed to have numerical index 1, unless it is finite-dimensional.