International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 890598, 10 pages
http://dx.doi.org/10.1155/2011/890598
Research Article

On Simultaneous Farthest Points in 𝐿 ∞ ( 𝐼 , 𝑋 )

Sh. Al-Sharif1 and M. Rawashdeh2

1Mathematics Department, Yarmouk University, Irbid 21163, Jordan
2Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid 22110, Jordan

Received 20 December 2010; Accepted 23 May 2011

Academic Editor: Siegfried Gottwald

Copyright © 2011 Sh. Al-Sharif and M. Rawashdeh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑋 be a Banach space and let 𝐺 be a closed bounded subset of 𝑋 . For ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑚 ) ∈ 𝑋 𝑚 , we set   𝜌 ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑚 , 𝐺 ) = s u p { m a x 1 ≤ 𝑖 ≤ 𝑚 ‖ 𝑥 𝑖 − 𝑦 ‖ ∶ 𝑦 ∈ 𝐺 } . The set 𝐺 is called simultaneously remotal if, for any ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑚 ) ∈ 𝑋 𝑚 , there exists 𝑔 ∈ 𝐺 such that   𝜌 ( 𝑥 1 , 𝑥 2 , … , 𝑥 𝑚 , 𝐺 ) = m a x 1 ≤ 𝑖 ≤ 𝑚 ‖ 𝑥 𝑖 − 𝑔 ‖ . In this paper, we show that if 𝐺 is separable simultaneously remotal in 𝑋 , then the set of ∞ -Bochner integrable functions, 𝐿 ∞ ( 𝐼 , 𝐺 ) , is simultaneously remotal in 𝐿 ∞ ( 𝐼 , 𝑋 ) . Some other results are presented.