Boundaries for strong Schur spaces (Q2922456)

Consider a real Banach space \(X\). A subset \(B\) of the dual unit sphere is said to be a boundary for \(X\) if every element of \(X\) attains its norm at some functional from \(B\).NEWLINENEWLINE \textit{V. P. Fonf} [Math. Notes 45, No. 6, 488--494 (1989); translation from Mat. Zametki 45, No. 6, 83--92 (1989; Zbl 0699.46010)] proved the following theorem: If \(X\) does not contain an isomorphic copy of \(c_0\), \(B\) is a boundary for \(X\) and \((B_n)_{n\in \mathbb{N}}\) is an increasing sequence of sets such that \(B=\bigcup_{n=1}^{\infty}B_n\), then there exist \(m\in \mathbb{N}\) and \(r>0\) such that \(\overline{\text{co}}^*(\pm B_m)\), the weak*-closed convex hull of \(B_m\cup -B_m\), contains \(rB_{X^*}\) (where \(B_{X^*}\) denotes the unit ball of \(X^*\)).NEWLINENEWLINE A Banach space \(X\) is said to be a strong Schur space if there exists a constant \(\eta>0\) such that for every \(\varepsilon>0\) every bounded \(\varepsilon\)-separated sequence in \(X\) contains a subsequence which is \(\eta\varepsilon\)-equivalent to the canonical basis of \(\ell^1\).NEWLINENEWLINE In this paper, the authors prove the following characterisation of strong Schur spaces in the spirit of the above mentioned theorem from [loc. cit.]. More precisely, they show that \(X\) is a strong Schur space if and only if there exists some \(\delta>0\) such that for every \(\alpha\in (0,\delta)\), every boundary \(B\) for \(X\) and every increasing sequence \((B_n)_{n\in \mathbb{N}}\) of sets with \(B=\bigcup_{n=1}^{\infty}B_n\), one can find a finite-codimensional subspace \(Z\subseteq X\) and an index \(m\in \mathbb{N}\) satisfying \(\alpha B_{Z^*}\subseteq \overline{\text{co}}^*(\pm B_m\cap Z)\).