Geometry of Banach spaces with property \(\beta\) (Q1806266)

Let \((X,\|\cdot\|)\) be a Banach space over the reals with dual space \(X^*\). \(X\) is said to have property \(\beta\) if there exists a system \(\{x_i,x^*_i\}_{i\in I}\subset X\times X^*\) and a real number \(0\leq\varepsilon< 1\) satisfying \(x^*_i(x_i)= 1=\|x_i\|= \|x^*_i\|\), \(|x^*_i(x_j)|\leq \varepsilon\), \(i\neq j\), \(\|x\|= \sup_{i\in I}|x^*_i(x)|\). The norm \(\|\cdot\|\) on \(X\) is said to be strongly differentiable at \(x\) if the one sided limit \(\lim_{t\to 0^+}{1\over t} (\|x+ th\|-\|x\|)\) exists uniformly on \(h\in S(X)\) -- the unit sphere of \(X\). In this note, the authors are concerned with the geometry of spaces with property \(\beta\). They prove that every Banach space can be isometrically and 1-complementably embedded into a Banach space which satisfies property \(\beta\) and has the same character of density. Then they show that property \(\beta\) satisfies a hereditary property. They study strong subdifferentiability of norms with property \(\beta\) to characterize separable polyhedral Banach spaces as those admitting a strongly subdifferentiable \(\beta\) norm. Finally, they provide examples of non-reflexive spaces whose usual norm fails property \(\beta\) and yet it can be approximated by norms with this property, namely \((L_1[0,1],\|\cdot\|_1)\) and \((C(K),\|\cdot\|_\infty)\) where \(K\) is a separable compact Hausdorff space.