Geometry of product spaces
Given a nonempty set \(S\), we denote by \(\mathbb R ^ S\) the set of all real valued functions on \(S\). Let \( \| \cdot \|_E\) be a norm on a linear subspace \(E\) of \(\mathbb R ^S\). We say that the normed space \(E\) is a full function space over \(S\) if, for any \(f \in E\) and \(g: S \rightarrow\mathbb R\) such that \(|g(s)| \leq |f(s)|\) for all \(s \in S\), it is true that \(g \in E\) and \(\| g \|_E \leq \| f \|_E\). Suppose that \(E\) is a full function space over a nonempty set \(S\) and \(\{ X_s \}_{s \in S}\) is a collection of normed spaces. For \(x \in \prod_{s \in S} X_s\), let \(E_x(s) = \| x(s)\|\) for all \(s \in S\). If \(E_x \in E\), let \(\|x\| = \|E_x\|_E\). This function defined on \(\{ x \in \prod_{s\in S} X_s : E_x \in E\}\) is a norm. The resulting normed space is said to be the \(E\)-direct sum of \(\{X_s\}_{s\in S}\) and is denoted by \(\mathcal{P}_E\{X_s\}_{s\in S}\). The standard \(p\)-direct sum (\(1\leq p\leq \infty\)) of normed spaces is an example of an \(E\)-direct sum with \(E = \ell_p\). The author studies the inheritance of some rotundity properties. Let \(X\) be a normed space with unit sphere \(S_X\). We denote the linear and convex spans of a subset \( A\) of \(X\) by \(\text{sp}\, A\) and \(\text{co}\, A\), respectively. Let \(k \in \mathbb N\). Then \(X\) is said to be \(k\)-rotund if, for any affinely independent \(x_1, \dotsc, x_{k+1} \in S_X\) \[ \frac{1}{k+1} \left\|\sum_{i=1}^{k+1} x_i \right\| < 1.\] \(X\) is said to be \(k\)-uniformly rotund if, for every \(\varepsilon > 0\), \[ \inf \left\{ 1-\frac{1}{k+1} \left\|\sum_{i=1}^{k+1} x_i \right\|: x_1, \dotsc, x_{k+1} \in S_X,\ V(x_1, \dotsc, x_{k+1}) \geq \varepsilon \right\} > 0,\] where \(\frac{1}{k!} V (x_1, \dotsc, x_{k+1})\) is the \(k\)-dimensional volume of \(\operatorname{co}\{x_1, \dotsc, x_{k+1}\}\). Then \(X\) is said to be \(k\)-uniformly rotund relative to every \(k\)-dimensional subspace if, for every \(k\)-dimensional subspace \(Y\) of \(X\) and every \(\varepsilon > 0\), \[ \inf \left\{ 1-\frac{1}{k+1} \left\|\sum_{i=1}^{k+1} x_i \right\|: x_1, \dotsc, x_{k+1} \in S_X,\ V(x_1, \dotsc, x_{k+1}) \geq \varepsilon,\ \operatorname{sp} \{x_1-x_{k+1}, \dotsc, x_k-x_{k+1}\} =Y \right\} > 0.\] The author obtains necessary and sufficient conditions for a product space to have one of the aforementioned rotundity properties.
$k$-uniform rotundity relative to every $k$-dimensional subspace
scientific article; zbMATH DE number 7370494
scientific article