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Convexification of super weakly compact sets and measure of super weak noncompactness. (English) Zbl 1471.46005

Let \(A\) be a subset of a Banach space \(X\), and let \(\textrm{co}(A)\) and \(\textrm{aff}(A)\) denote the convex hull and the affine hull of \(A\). We say that a subset \(B\) of a Banach space \(Y\) is finitely representable in \(A\) if for every finite subset \(B_0\) of \(B\) and \(r>1\) there is a finite subset \(A_0\) of \(A\) and an affine isomorphism \(T:\textrm{aff}(B_0)\to\textrm{aff}(A_0)\) such that \(T(\textrm{co}(B_0))=\textrm{co}(A_0)\) and \(r^{-1}\|x-y\|\leq \|Tx-Ty\|\leq r\|x-y\|\) for all \(x,y\in\textrm{aff}(B_0)\).
We say that the set \(A\) is relatively super weakly compact if every subset finitely representable in \(A\) is relatively weakly compact. In [Stud. Math. 199, No. 2, 145–169 (2010; Zbl 1252.46009); J. Convex Anal. 25, No. 3, 899–926 (2018; Zbl 1408.46014)], L.-X. Cheng et al. studied this property for convex bounded subsets, and asked if the closed convex hull of a relatively super weakly compact subset inherits the property. In this paper the author gives a positive answer by introducing a quantity \(\sigma(A)\) whose properties are similar to that of the Hausdorff measure of non-compactness; in particular, for a bounded subset \(A\) we have \[\sigma(A)=\sigma(\textrm{co}(A))=\sigma(\overline{A}^w),\] where \(\overline{A}^w\) denotes the weak closure, and \(\sigma(A)=0\) if and only if \(A\) is relatively super weakly compact.
The author also proves a fixed point theorem for \(\sigma\)-condensing maps.

MSC:

46A50 Compactness in topological linear spaces; angelic spaces, etc.
46B07 Local theory of Banach spaces
46B50 Compactness in Banach (or normed) spaces
47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10 Fixed-point theorems
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References:

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