A metric space (X, d) is called an M-space if for every x and y in X and for every r 6 [0, A] we have B[x, r] Cl B[y, A — r] = {2} for some z € X, where A = d(x, y). It is the object of this paper to study M-spaces in terms of proximinality properties of certain sets. 0. Introduction. Let (X, d) be a metric space, and G be a closed subset of X. For x E X, let p(x,G) = inf{d(x, y) : y E G}. If t...

We derive transitivity of various degrees of proximinality in Banach spaces. When the transitivity does not carry forward to the bigger space we investigate these properties under some additional assumptions of the intermediate space. For instance, we show 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 stro...

In this paper, we show that if E is an order continuous Köthe function space and Y is a separable subspace ofX, then E(Y ) is ball proximinal in E(X) if and only if Y is ball proximinal in X. As a consequence, E(Y ) is proximinal in E(X) if and only if Y is proximinal in X. This solves an open problem of Bandyopadhyay, Lin and Rao. It is also shown that if E is a Banach lattice with a 1-uncondi...