Almost squareness and strong diameter two property in tensor product spaces (Q2293246)

In [\textit{T. A. Abrahamsen} et al., J. Math. Anal. Appl. 434, No. 2, 1549--1565 (2016; Zbl 1335.46006)], almost square Banach spaces were introduced and studied. A Banach space \(X\) is called almost square (ASQ) if, for every finite set of points \(x_1,\dots,x_n\in X\) with norm one and for every \(\varepsilon>0\), there is a \(y\in X\) with norm one such that \(\|x_i+y\|\leq 1+\varepsilon\) for all \(i\in\{1,\dots,n\}\). The prototype of an almost square space is \(c_0\). It turns out that every almost square space has the strong diameter~\(2\) property (SD2P), that is, every finite convex combination of slices of its unit ball has diameter \(2\). In [\textit{J. Langemets} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 111, No. 3, 841--853 (2017; Zbl 1385.46007)], it was shown that the injective tensor product of an ASQ space with any nontrivial space is ASQ. The main result of the paper under review proves that the projective tensor product of two ASQ spaces is ASQ, too. This result provides a large class of nontrivial examples of ASQ projective tensor product spaces. The author also introduces a strengthening of the SD2P, called the sequential SD2P, and shows that, if a Banach space \(X\) has the sequential SD2P, then every projective symmetric tensor product of \(X\) has the SD2P.