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Sphere equivalence, Property H, and Banach expanders

100%
Cheng Q.
Studia Mathematica
|
2016
|
tom 233
|
nr 1
67-83
EN
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.
2
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Characterization of normed linear spaces with generalized Mazur intersection property

64%
Dong Y., Cheng Q.
Studia Mathematica
|
2013
|
tom 219
|
nr 3
193-200
EN
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.
3
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On some new characterizations of weakly compact sets in Banach spaces

51%
Cheng L., Cheng Q., Luo Z.
Studia Mathematica
|
2010
|
tom 201
|
nr 2
155-166
EN
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 $p ≥ sup_{C}$ such that p² is everywhere Fréchet differentiable in X*; and as a consequence, the space X is a weakly compactly generated space if and only if there exists a continuous and w*-l.s.c. Fréchet smooth (not necessarily equivalent) norm on X*.
4
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Minimal ball-coverings in Banach spaces and their application

51%
Cheng L., Cheng Q., Shi H.
Studia Mathematica
|
2009
|
tom 192
|
nr 1
15-27
EN
By a ball-covering 𝓑 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 𝓑 is called minimal if its cardinality $𝓑^{#}$ 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 characterization of superreflexive spaces in terms of ball-coverings. Finally, we show that every infinite-dimensional Banach space admits an equivalent norm such that there is an infinite-dimensional quotient space possessing a countable ball-covering.
5
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On super-weakly compact sets and uniformly convexifiable sets

51%
Cheng L., Cheng Q., Wang B., Zhang W.
Studia Mathematica
|
2010
|
tom 199
|
nr 2
145-169
EN
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 theorem, Enflo's renorming technique, Grothendieck's lemma and the Davis-Figiel-Johnson-Pełczyński lemma.
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