On the farthest points in convex metric spaces and linear metric spaces

We prove some results on the farthest points in convex metric spaces and in linear metric spaces. The continuity of the farthest point map and characterization of strictly convex linear metric spaces in terms of farthest points are also discussed.

Monotonicity and the Dominated Farthest Points Problem in Banach Lattice

We introduce the dominated farthest points problem in Banach lattices. We prove that for two equivalent norms such thatXbecomes an STM and LLUM space the dominated farthest points problem has the same solution. We give some conditions such that under these conditions the Fréchet differentiability of the farthest point map is equivalent to the continuity of metric antiprojection in the dominated farthest points problem. Also we prove that these conditions are equivalent to strong solvability of the dominated farthest points problem. We prove these results in STM, reflexive STM, and UM spaces. Moreover, we give some applications of the stated results in Musielak-Orlicz spacesL ​ϕ(μ)andE ​ϕ(μ)over nonatomic measure spaces in terms of the functionϕ. We will prove that the Fréchet differentiability of the farthest point map and the conditionsϕ∈Δ2andϕ>0in reflexive Musielak-Orlicz function spaces are equivalent.

Convexities and Existence of the Farthest Point

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