Uniform convexity properties of norms on a super-reflexive Banach space
As shown by \textit{G. Pisier} [Isr. J. Math. 20, 326-350 (1975; Zbl 0344.46030)], on every super-reflexive Banach space X there exists a uniformly convex equivalent norm with a modulus of convexity of power- type. The present author asks now: ''What can be said of any equivalent norm on a super-reflexive Banach space?'' The answer is that every equivalent norm has some uniform convexity and smoothness properties. The results are given in two statements, in Theorem 3 and in Proposition 6, which form the main content of Section I. For Theorem 3 the author introduces a new notion \(\Delta_{\| \|}(x,\epsilon)\), called ''the modulus of strong extremality at the point \(x\in S(X)''\), S(X) being the unit sphere of X. Theorem 3 states that for any N and for all \(\eta\) : \(0<\eta <1\) there exists \(K(\eta)>0\) such that the relation \(B_ N(X)\subseteq conv[\Omega_ N(K(\eta),q)]+\eta B_ N(X)\) holds. Here \(B_ N(X)\) means the unit ball of X and conv[ ] the convex hull of the set \(\Omega (K,q)=\{x\in S(X)| \Delta_ N(x,\epsilon)\geq K\epsilon^ q,\forall \epsilon >0\}.\) For Proposition 6 the author also introduces a new notion called ''\(\phi\)- strongly exposed point \(x\in C''\), \(C\) being a subset of \(X\). \(\phi\) means an increasing function in \([0,1]\) such that \(\phi (0)=0\). Proposition 6 states that for any equivalent norm N and for all \(\eta\), \(0<\eta <1\), there exists a function \(\phi\eta\) and a constant \(K(\eta)>0\) such that the relation \(B_ N(X)\subseteq conv\{{\mathcal E}_ N(\phi \eta)\cap \Omega_ N(K(\eta),q)\}+\eta B_ N(X)\) holds. Here \({\mathcal E}_ N(\phi \eta)\) means the set of the \(\phi\eta\)-strongly exposed points in \(B_ N(X).\) Section II is devoted to applications of the results obtained in Section I. There are given the statements: 1) Every super-reflexive space is almost uniformly smooth (a.u.s.). (A.u.s. being a notion introduced by the author in her earlier papers.) 2) If the super-reflexive Banach space \(X\) is quasi-transitive (in a given sense), then \(X\) is uniformly convex with a modulus of convexity of power-type. The uniform approximation property is discussed.
scientific article; zbMATH DE number 3973578
scientific article