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This paper presents two counterexamples about ball-coverings of Banach spaces and shows a new characterization of uniformly non-square Banach spaces via ball-coverings. [ABSTRACT FROM AUTHOR]

Based on an application of the Davis-Figiel-Johnson-Pelzyśki procedure, this note shows that every weakly compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell-Schlumprecht and Hájek-Johanis can be extended and localized to weakly compact convex subsets of an arbitrary Banach space. [ABSTRACT FROM AUTHOR]

To guarantee every real-valued convex function bounded above on a set is continuous, how ”thick” should the set be? For a symmetric set A in a Banach space E, the answer of this paper is: Every real-valued convex function bounded above on A is continuous on E if and only if the following two conditions hold: i) spanA has finite co-dimentions and ii) co A has nonempty relative interior. This paper also shows that a subset A ⊂ E satisfying every real-valued convex function bounded above on A is continuous on E if (and only if) every real-valued linear functional bounded above on A is continuous on E, which is also equivalent to that every real-valued convex function bounded on A is continuous on E. [ABSTRACT FROM AUTHOR]