Some Equivalent Conditions of Proximinality in Nonreflexive Banach Spaces

In this paper, we discuss the relation between τ-strongly Chebyshev, approximatively τ-compact k-Chebyshev, approximatively τ-compact, τ-strongly proximinal and proximinal sets, where τ is the norm or the weak topology. We give some equivalent conditions regarding the above proximinality. Furthermore, we also propose the necessary and sufficient conditions that a half-space is τ-strongly proximinal, approximatively τ-compact and τ-strongly Chebyshev.