Pradipta Bandyopadhyay

Farthest points and the farthest distance map

In this paper, we consider farthest points and the farthest distance map of a closed bounded set in a Banach space. We show, inter alia, that a strictly convex Banach space has the Mazur-like intersection property for weakly compact sets if and only if every such set is the closed convex hull of its farthest points, and recapture a classical result of Lau in a broader set-up. We obtain an expression for the subdifferential of the farthest distance map in the spirit of Preiss’ Theorem which in turn extends a recent result of Westphal and Schwartz, showing that the subdifferential of the farthest distance map is the unique maximal monotone extension of a densely defined monotone operator involving the duality map and the farthest point map.

On a new asymptotic norming property

Abstract In this work, we introduce a new Asymptotic Norming Property (ANP) which lies between the strongest and weakest of the existing ones, and obtain isometric characterisation of it. The corresponding w∗-ANP turns out to be equivalent on the one hand, to Property (V) introduced by Sullivan, and to a ball separation property on the other. We also study stability properties of this new ANP and its w∗-version.

Densely ball remotal subspaces of C(K)

We call a subspace Y of a Banach space X a DBR subspace if its unit ball BY admits farthest points from a dense set of points of X. In this paper, we study DBR subspaces of C(K). In the process, we study boundaries, in particular, the Choquet boundary of any general subspace of C(K). An infinite compact Hausdorff space K has no isolated point if and only if any finite co-dimensional subspace, in particular, any hyperplane is DBR in C(K). As a consequence, we show that a Banach space X is reflexive if and only if X is a DBR subspace of any superspace. As applications, we prove that any M-ideal or any closed ∗-subalgebra of C(K) is a DBR subspace of C(K). It follows that C(K) is ball remotal in C(K)∗∗.