Characterizations of simultaneous farthest point in normed linear spaces with applications

In this paper, we consider a problem of best approximation (simultaneous farthest point) for bounded sets in a real normed linear space X. We study simultaneous farthest point in X by elements of bounded sets, and present various characterizations of simultaneous farthest point of elements by bounded sets in terms of the extremal points of the closed unit ball of X ^*, where X ^* is the dual space of X. We establish the characterizations of simultaneous farthest points for bounded sets in , the space of all real-valued continuous functions on a compact topological space Q endowed with the usual operations and with the norm . It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, [9, Theorem 1.13]) is a technical method to represent the distance from a bounded set to a compact convex set in X which specifically concentrates on the Hahn-Banach Theorem in X.