Orlicz spaces which are noncreasy (Q1606081)

Let \(X\) be a real Banach space, \(S(X)\) its unit sphere. Denote \(S(x^*)=\{x\in X\mid \|x\|\leq 1\) and \(x^*(x)\geq 1-d\}\). Following S. Prus, a Banach space is said to be noncreasy if \(\text{diam} (S(x^*,0)\cap S(y^*,0))=0\) for any two different functionals \(x^*,y^*\in S(X^*)\); a Banach space is uniformly noncreasy if, for any \(\varepsilon>0 \), there exists \(\delta>0\) such that \(\text{diam}(S(x^*,\delta)\cap S(y^*,\delta))\leq \varepsilon\) for any \(x^*,y^*\in S(X^*)\), \(\|x^*-y^*\|\geq \varepsilon\). It is proved that a Banach space is noncreasy whenever it is rotund or smooth, equivalently, if \(X^*\) is weakly star noncreasy. Also, if \(X\) is a Banach space, then \(\delta (x,\varepsilon)>0\) for any \(x\in S(X)\) and \(\varepsilon>0\) if and only if \(X\) is midpoint locally uniformly rotund. As a corollary it follows that if a Banach space is uniformly noncreasy, then any \(x\in S(X)\) is a strongly extreme point or a Fréchet differentiable point. Further, criteria for noncreasiness of Orlicz finctional spaces are obtained.