On geometry of Banach spaces with property \(\alpha\) (Q1922925)

A real Banach space \(X\) is said to have property \(\alpha\), provided there exist a system \((x_i,x^*_i)\in X\times X^*\), \(i\in I\), and \(\varepsilon\), \(0\leq\varepsilon<1\), such that: \(x^*_i(x_i)= 1=|x_i|=|x^*_i|\), \(|x^*_i(x_j)|\leq\varepsilon\); and \(B_X=\text{cl conv}\{\pm x_i:i\in I\}\), (\(B_X\) -- the closed unit ball of \(X\)). This notion was introduced by \textit{J. Lindenstrauss} [Isr. J. Math. 1, 139-148 (1963; Zbl 0127.06704)]in connection with the Bishop-Phelps property for operators (the density of norm attaining operators in the unit ball). \textit{W. Schachermayer} [Isr. J. Math. 44, 201-212 (1983; Zbl 0542.46013)]proved that property \(\alpha\) is not hereditary, i.e. it is not inherited by every closed subspace. However, a kind of heredity holds: If \(X\) has property \(\alpha\) then for every subspace \(Y\) of \(X\), there exists a subspace \(Z:Y\subset Z\subset X\), with the same density character as \(Y\) and having property \(\alpha\) (Theorem 2.1). The second part of the paper is concerned with property (H), meaning that every sequence in \(S_X\) (the unit sphere of \(X\)), converging weakly to an element of \(S_X\), is norm-convergent. Replacing the word ``sequence'' by ``net'' one obtains the Kadec-Klee property. A Banach space with properties \(\alpha\) and (H) is not reflexive (Theorem 3.2), and every Banach space with property (H) has no weakly uniformly rotund point (Corollary 3.7). The paper ends with some open problems.