In this paper, we consider the concepts farthest points and nearest points in normed linear spaces, We obtain a necessary and coecient conditions for proximinal, Chebyshev, remotal and uniquely remotal subsets in normed linear spaces. Also, we consider -remotality, -proximinality, coproximinality and co-remotality.

Let (S,Σ , μ) be a complete positive σ -finite measure space and let X be a Banach space. We are concerned with the proximinality problem for the best simultaneous approximations to two functions in L p(S,Σ , X). Let Σ0 be a sub-σ -algebra of Σ and Y a nonempty locally weakly compact convex subset of X such that span Y and its dual have the Radon–Nikodym property. We prove that L p(S,Σ0, Y ) is...

We investigate a variation of the transitivity problem for proximinality properties of subspaces and intersection properties of balls in Banach spaces. For instance, we prove that if Z ⊆ Y ⊆ X, where Z is a finite co-dimensional subspace of X which is strongly proximinal in Y and Y is an M -ideal in X, then Z is strongly proximinal in X. Towards this, we prove that a finite co-dimensional proxi...

In this paper, we show that a closed convex set C of a Banach space is strongly proximinal (proximinal, resp.) in every Banach space isometrically containing it if and only if C is locally (weakly, resp.) compact. As a consequence, it is proved that local compactness of C is also equivalent to that for every Banach space Y isometrically containing it, the metric projection from Y to C is nonemp...