Diameter 2 properties and convexity (Q2809348)

Let us start by recalling some notions.NEWLINENEWLINEA Banach space has the local diameter 2 property (LD2P), respectively diameter 2 property (D2P), strong diameter 2 property (SD2P), if every slice, respectively nonempty relatively weakly open subset, every convex combination of slices, of its unit ball has diameter 2. It is known that SD2P \(\Rightarrow\) D2P and D2P \(\Rightarrow\) LD2P, and in general these implications are strict.NEWLINENEWLINEA Banach space \(X\) is midpoint locally uniformly rotund (MLUR), respectively weakly midpoint locally uniformly rotund (\(w\)-MLUR) if every \(x\) in the unit sphere \(S_X\) of \(X\) is a strongly extreme point, respectively strongly extreme point in the weak topology.NEWLINENEWLINEThe authors start by observing that if a Banach space is \(w\)-MLUR, then the LD2P is equivalent to the D2P. They show that the Banach space \(c_0\) can be renormed in such a way that it is \(w\)-MLUR and has the SD2P. The authors say that a Banach space \(X\) has the local diameter 2 property+ (LD2P+) if for every \(\varepsilon>0\), every slice \(S\) of \(B_X\), and every \(x\in S\cap S_X\) there exists \(y\in S\) such that \(\|x-y\|>2-\varepsilon\). (Note that LD2P+ is the same property as what was named \textit{spaces with bad projections} in [\textit{Y. Ivakhno} and \textit{V. Kadets}, Visn. Khark. Univ., Ser. Mat. Prykl. Mat. Mekh. 645, No. 54, 30--35 (2004; Zbl 1071.46015)] and what was named \textit{diametral local diameter 2 property} in [\textit{J. Becerra Guerrero}, \textit{G. López Pérez} and \textit{A. Rueda Zoca}, ``Diametral diameter two properties in Banach spaces'', \url{arXiv:1509.02061}]) Obviously, LD2P+ \(\Rightarrow\) LD2P. On the other hand, the Banach space \(c_0\) even has the SD2P, but fails the LD2P+. The main result of the paper under review says that the Banach space \(C[0,1]\) can be equivalently renormed in such a way that it is MLUR and has both D2P and LD2P+ simultaneously. From here, the authors go on to construct a Banach space which is MLUR, has the D2P, the LD2P+, and has convex combinations of slices of arbitrarily small diameter.NEWLINENEWLINEThe paper ends with some interesting open questions. To point out just two: {\parindent=8mm \begin{itemize}\item[(1)] Does there exist an equivalent MLUR norm on \(c_0\) with the LD2P? \item[(2)] Does there exist a Banach space with the LD2P+ which fails the D2P?NEWLINENEWLINE\end{itemize}}