On the extension of \(k\)-NUC norms (Q5944180)

The reviewed article continues investigations of extension of norms, which have some rotundity or convexity properties, that was started with the paper of \textit{K. John} and \textit{V. Zizler} [Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 24, 703-707 (1976; Zbl 0344.46048)] and \textit{D. Kutzarova} [J. Math. Anal. Appl. 162, No. 2, 322-328 (1976; Zbl 0757.46026)] introduced the notion of \(k\)-nearly uniformly rotund (shortly: \(k\)-NUR) Banach space \(X\), which is called \(k\)-NUR (\(k\) is a fixed natural number), if for every \(\varepsilon >0\) there exists \(\delta >0\) such that for every sequence \(( x_{n}) \) in \(X\) with \(\|x_{n}\|=1\) and \(\inf \|x_{n}-x_{i}\|\geq \varepsilon \) (\(n\neq i\)) there are indices \(( n_{j}) _{j=1}^{k}\) such that \(\|\sum\nolimits_{j=1}^{k}x_{n_{j}}\|\leq ( 1+\delta) k\).NEWLINENEWLINENEWLINEAuthors proved that if \(Y\) is a subspace of a Banach space \(X\) and both spaces \(X\) and \(Y\) admit \(k\) -NUR norms, say \(\|\cdot \|_{X}\) and \(|\cdot |_{Y}\) \ respectively, then \(X\) admits such \(k\)-NUR norm \(|\cdot |_{X}\), which restriction to \(Y\) is coincides with \(|\cdot |_{Y}\).