Midpoint locally uniform rotundity and a decomposition method for renorming (Q2730925)

Let \(A\) be a subset of a normed linear space \(X\). For \(\varepsilon> 0\) a point \(a\in A\) is said to be an \(\varepsilon\)-strongly extreme point of \(A\) if there exists \(\delta> 0\) such that \(u,v\in A\) and \(\|a-(u+ v)/2\|< \delta\) implies \(\|u-v\|< \varepsilon\).NEWLINENEWLINENEWLINEThe normed space \(X\) is midpoint locally uniformly rotund (MLUR) if, for all \(\varepsilon> 0\), all the points of its unit sphere are \(\varepsilon\)-strongly extreme points of its unit ball.NEWLINENEWLINENEWLINEThis paper deals with MLUR renorming of normed spaces. The main result is that such a MLUR renorming of \(X\) exists if and only if, for every \(\varepsilon> 0\), \(X\) can be decomposed as \(X= \bigcup^\infty_{n=1} X_{n,\varepsilon}\) such that all the points of \(X_{n,\varepsilon}\) are \(\varepsilon\)-extreme points of the convex hull of \(X_{n,\varepsilon}\). A similar characterization for the existence of LUR renormings has been obtained by the three first authors in [Proc. Lond. Math. Soc., III. Ser. 75, No. 3, 619-640 (1997; Zbl 0909.46011)].NEWLINENEWLINENEWLINEThe authors show how these characterizations can be used to get another proof of results on LUR and MLUR renorming obtained by \textit{R. Haydon} [Proc. Lond. Math. Soc., III. Ser. 78, No. 3, 541-584 (1999; Zbl 1036.46003)]. They also give an explicit renorming of the James space \(J\) such that the bidual \(J^{**}\) is MLUR.