Compactly uniformly convex spaces and property \((\beta)\) of Rolewicz (Q1947292)

The property \((\beta)\) of \textit{S. Rolewicz} [Stud. Math. 85, No. 1, 27--35 (1986; Zbl 0642.46011); ibid. 87, No. 2, 181--191 (1987; Zbl 0652.46010)] is a property a little weaker than the classical concept of uniform convexity, and a little stronger than nearly uniform convexity. Recently, the last two authors of the paper under review introduced the concept of compact uniform convexity [\textit{J. P. Revalski} and \textit{N. V. Zhivkov}, J. Convex Anal. 19, No.~4, 1153--1166 (2012; Zbl 1276.41029)]. In this rather technical, but well-written and interesting, paper, the authors prove in Section~2 that property \((\beta)\) and compact uniform convexity is one and the same property (Proposition 2.5). In a short Section 3, the authors characterize property \((\beta)\) in terms of families of locally finite graphs. In Section~4, new examples of spaces with property \((\beta)\) are produced by looking at compact and nuclear operators between \(\ell_p\)-spaces. One result is that if \(1<q<2<p<\infty\), then the space of (compact) operators from \(\ell_p\) into \(\ell_q\) has property \((\beta)\). On the other hand, in the complementary ranges of \(p\) and \(q\), this space does not have property \((\beta)\) in any equivalent norm. In the final Section~5, an estimate for the \((\beta)\)-modulus is obtained, provided that the moduli of nearly uniform convexity and of nearly uniform smoothness are of certain power types.