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 \textit{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 \textit{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)], \textit{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.