The aim of this paper is to give several well-known results of Banach spaces possessing, the Radon-Nikodym property. It is interesting to see how this property can be described in terms of Asplund spaces, spaces admitting KKP, and rotund norm. We study nearly about an open problem [1] mentioned below.

In this paper, we present the basic facts about vasak (WCD) space with application to the construction of equivalent norm on the vasak spaces. We study nearly about an open problem raised in [1].

In this paper, we give some new conditions for the existence and uniqueness of mutually nearest points of two sets, i.e., two points which achieve the minimum distance between two sets in a metric or Banach spaces. These conditions are seen in case of compact sets, weak Compact sets, closed and convex sets, weakly sequentially compact sets and boundedly compact sets and their combinations. The study is confined to metric spaces and normed or Banach spaces. Some geometric properties of a Banach spaces like; strictly convexity, uniformly convexity, P-Property and weak P-Property are introduced. Also, we introduce the concept of generalized weak P-Property and give some interesting results. The present work may be briefly outlined as follows: It is the mathematical study that is motivated by the desire to seek answers to the following basic questions, among others. Which subsets are mutually proximinal? How does one recognize when given elements x ∈ A and y ∈ B are the nearest points of A and B? which is called a natural extension of the best approximation problem. Can one describe some useful algorithms for actually computing nearest points between two given sets? And how to find closely related sets to the proximity maps.