On simultaneous farthest points in \(L^{\infty}(I, X)\)
Summary: Let \(X\) be a Banach space and let \(G\) be a closed bounded subset of \(X\). For \((x_1, x_2, \dots, x_m) \in X^m\), we set \(\rho(x_1, x_2, \dots, x_m, G) = \text{sup} \{\text{max}_{1\leq i\leq m} \| x_i - y\| : y \in G\}\). The set \(G\) is called simultaneously remotal if, for any \((x_1, x_2, \dots, x_m) \in X^m\), there exists \(g \in G\) such that \(\rho (x_1, x_2, \dots, x_m, G) = \text{max}_{1\leq i\leq m} \| x_i - g\|\). In this paper, we show that if \(G\) is separable simultaneously remotal in \(X\), then the set of \(L^{\infty}\)-Bochner integrable functions, \(L^{\infty}(I, G)\), is simultaneously remotal in \(L^{\infty}(I, X)\). Some other results are presented.
scientific article; zbMATH DE number 5946489