A generalization of uniformly extremely convex Banach spaces (Q288783)

Let \(X\) be a real Banach space and \(k\in \mathbb{N}\). The authors introduce the following notion: \(X\) is said to be \(k\)-uniformly extremely convex, if for all sequences \((x_n^{(1)})_{n\in \mathbb{N}},\dots,(x_n^{(k+1)})_{n\in \mathbb{N}}\) in the unit sphere of \(X\) such that there exists a norm-one functional \(f\) on \(X\) with \(\lim_{n\to \infty}f(x_n^{(i)})=1\) for all \(i=1,\dots,k+1\), one has \(V(x_n^{(1)},\dots,x_n^{(k+1)})\to 1\).NEWLINENEWLINE Here, for given points \(y_1,\dots,y_{k+1}\) on the unit sphere of \(X\), \(V(y_1,\dots,y_{k+1})\) stands for the volume enclosed by \(y_1,\dots,y_{k+1}\), i.e., the supremum over all norm-one functionals \(f_1,\dots,f_k\) on \(X\) over the determinants of the \((k+1)\times (k+1)\)-matrices whose first row consists of \(1\)s and whose \(i\)-th row is given by \((f_{i-1}(y_1),\dots,f_{i-1}(y_{k+1}))\) for \(i=2,\dots,k+1\).NEWLINENEWLINE The \(1\)-uniformly extremely convex spaces are precisely the uniformly extremely convex spaces introduced by \textit{S. Wulede} and \textit{W. Ha} in [Proc. R. Soc. Edinb., Sect. A, Math. 142, No. 1, 215--224 (2012; Zbl 1256.46008)].NEWLINENEWLINE The authors study the relation of their newly introduced classes of spaces with other geometric properties of Banach spaces and provide different characterisations and counterexamples. For instance, they show that \(X\) is \(k\)-uniformly extremely convex if and only if \(X\) is \(k\)-strictly convex and has the drop property. They also provide a characterisation of \(k\)-uniformly extremely convex spaces in terms of reflexivity and an additional condition on the sets \(\{x\in X:\|x\|=1=f(x)\}\) for norm-one functionals \(f\).