We study the uniform classification of the unit spheres of general Banach sequence spaces. In particular, we obtain some interesting applications involving Property H introduced by Kasparov and Yu, and Banach expanders.

Let 𝓐 be a compatible collection of bounded subsets in a normed linear space. We give a characterization of the following generalized Mazur intersection property: every closed convex set A ∈ 𝓐 is an intersection of balls.

We show several characterizations of weakly compact sets in Banach spaces. Given a bounded closed convex set C of a Banach space X, the following statements are equivalent: (i) C is weakly compact; (ii) C can be affinely uniformly embedded into a reflexive Banach space; (iii) there exists an equivalent norm on X which has the w2R-property on C; (iv) there is a continuous and w*-lower semicontinuous seminorm p on the dual X* with $p\ge su{p}_{C}$ such that p² is everywhere Fréchet differentiable in X*; and as a...

By a ball-covering of a Banach space X, we mean a collection of open balls off the origin in X and whose union contains the unit sphere of X; a ball-covering is called minimal if its cardinality ${}^{}$ is smallest among all ball-coverings of X. This article, through establishing a characterization for existence of a ball-covering in Banach spaces, shows that for every n ∈ ℕ with k ≤ n there exists an n-dimensional space admitting a minimal ball-covering of n + k balls. As an application, we give a new...

This paper mainly concerns the topological nature of uniformly convexifiable sets in general Banach spaces: A sufficient and necessary condition for a bounded closed convex set C of a Banach space X to be uniformly convexifiable (i.e. there exists an equivalent norm on X which is uniformly convex on C) is that the set C is super-weakly compact, which is defined using a generalization of finite representability. The proofs use appropriate versions of classical theorems, such as James' finite tree...

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