Convexities and Existence of the Farthest Point

Abstract EN

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.