Browsing by Subject "Super weak compactness"

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    On uniformly convex functions
    (Elsevier, 2022) Raja Baño, Matías; Grelier, Guillaume Guy Marcel; Matemáticas
    Non-convex functions that yet satisfy a condition of uniform convexity for non-close points can arise in discrete constructions. We prove that this sort of discrete uniform convexity is inherited by the convex envelope, which is the key to obtain other remarkable properties such as the coercivity. Our techniques allow to retrieve Enflo’s uniformly convex renorming of super-reflexive Banach spaces as the regularization of a raw function built from trees. Among other applications, we provide a sharp estimation of the distance of a given function to the set of differences of Lipschitz convex functions. Finally, we prove the equivalence of several possible ways to quantify the super weakly noncompactness of a convex subset of a Banach space.
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    Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness
    (Elsevier, 2023) Raja Baño, Matías; Grelier, Guillaume Guy Marcel; Matemáticas
    We introduce a measure of super weak noncompactness Γ defined for bounded subsets and bounded linear operators in Banach spaces that allows to state and prove a charac-terization of the Banach spaces which are subspaces of a Hilbert-generated space. The use of super weak compactness and Γcasts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.