Some remarks on generalised lush spaces (Q2787151)

A Banach space \(X\) has the Mazur-Ulam property (MUP) if, for every Banach space \(Y\), every surjective isometry \(T:S_X \rightarrow S_Y\) between the unit spheres can be extended to a linear isometry between \(X\) and \(Y\). In these terms, the famous Tingley problem asks whether every Banach space has the MUP. Recently, \textit{D.-N. Tan} et al. [Stud. Math. 219, No. 2, 139--153 (2013; Zbl 1296.46009)] introduced a new propery, called generalised lushness (GL in short), which implies the MUP. Namely, a real Banach space \(X\) is GL, if for every \(x \in S_X\) and every \(\varepsilon > 0\), there is a slice \(S= \{u \in B_X: x^*(u) > 1 - \varepsilon\}\) such that \(x \in S\) and, for every \(y \in S_X\), NEWLINE\[NEWLINE \mathrm{d}(y, S) + \mathrm{d}(y, - S) < 2 + \varepsilon. NEWLINE\]NEWLINE The paper under review is devoted to the study of this Banach space property. It is shown that GL is stable under ultraproducts and under passing to a large class of \(F\)-ideals, in particular to \(M\)-ideals. In this way, a technical mistake made in the original Tan, Huang, and Liu paper (see [loc.\,cit.]) is corrected. It is demonstrated that a close relative to GL passes from the bidual to the original space, and that GL of \(X^{**}\) implies the MUP. The unit sphere of a GL-space \(X\) of \(\dim X \geq 2\) does not possess LUR points.