Remarks on diameter 2 properties (Q2841074)

The authors say that a Banach space \(X\) has the local diameter~\(2\) property if every open slice of its unit ball \(B_X\) has diameter~\(2\), \(X\) has the diameter~\(2\) property if every non-empty relatively weakly open subset of \(B_X\) has diameter~\(2\), and \(X\) has the strong diameter~\(2\) property if every convex combination of open slices of \(B_X\) has diameter~\(2\). The strong diameter~\(2\) property implies the diameter~\(2\) property, and the diameter~\(2\) property obviously implies the local diameter~\(2\) property. The paper surveys known examples, presents new ones and investigates properties of spaces with the diameter~\(2\) property. The final section lists some open questions, in particular it is asked whether the three diameter~\(2\) properties actually coincide.NEWLINENEWLINEReviewer's remark: Recently it was shown that the diameter~\(2\) property does not imply the strong diameter~\(2\) property [\textit{M. D.~Acosta, J.~Becerro Guerrero} and \textit{G.~López Pérez}, ``Stability results on diameter two properties''; \textit{R.~Haller, J.~Langemets}, ``Two remarks on diameter~\(2\) properties''] and that the local diameter~\(2\) property does not imply the diameter~\(2\) property [\textit{J.~Becerra Guerrero, G.~López Pérez} and \textit{A.~Rueda Zoca}, ``Big slices versus big relatively weakly open subsets in Banach spaces'', \texttt{arXiv:1304.4397}].