Abstract and Applied Analysis
Volume 2011 (2011), Article ID 139597, 9 pages
http://dx.doi.org/10.1155/2011/139597
Research Article

Convexities and Existence of the Farthest Point

Z. H. Zhang and C. Y. Liu

College of Fundamental Studies, Shanghai University of Engineering Science, Shanghai 201600, China

Received 10 July 2011; Revised 20 September 2011; Accepted 20 September 2011

Academic Editor: Toka Diagana

Copyright © 2011 Z. H. Zhang and C. Y. Liu. 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

Five counterexamples are given, which show relations among the new convexities and some important convexities in Banach space. Under the assumption that Banach space 𝑋 is nearly very convex, we give a sufficient condition that bounded, weakly closed subset of 𝑋 has the farthest points. We also give a sufficient condition that the farthest point map is single valued in a residual subset of 𝑋 when 𝑋 is very convex.