On simultaneous farthest points in \(L^{\infty}(I, X)\)
type: Item
instance of: scholarly article
title: On simultaneous farthest points in \(L^{\infty}(I, X)\)
zbMATH Open document ID: 1235.46020
DOI: 10.1155/2011/890598
author: Juan-Miguel Gracia
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2011-09-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 46B20
Mathematics Subject Classification ID: 46E40
zbMATH DE Number: 5946489
zbMATH Keywords: Bochner integrable functions
zbMATH Keywords: simultaneously remotal
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/890598
OpenAlex ID: W2044741087
Wikidata QID: Q58687976
cites work: Remotal sets revisited
cites work: Farthest points in W*-compact sets
cites work: Farthest points of sets in uniformly convex Banach spaces
cites work: Farthest points in weakly compact sets
cites work: A remark on uniquely remotal sets in C(K,X)
cites work: Remotal sets in the space of \(p\)-integrable functions
cites work: Q3431279
cites work: REMARKS ON REMOTAL SETS IN VETOR VALUED FUNCTION SPACES
cites work: Characterizations of simultaneous farthest point in normed linear spaces with applications
cites work: Q4148368
description: scientific article; zbMATH DE number 5946489