REMOTALITY OF EXPOSED POINTS R. KHALIL 1 , S. HAYAJNEH, M. HAYAJNEH AND M. SABABHEH 2 Abstract. In this article, we discuss the problem of remotality of exposed points of bounded sets in certain Banach spaces. Indeed, we present a full characterization of a class of exposed points that are remotal points. 1. Introduction and preliminaries Let X be a Banach space, and E be a closed bounded convex subset of X . For x ∈ X , let D(x, E) = sup e∈E ‖x − e‖ be the maximum distance from x to E. If an e ∈ E exists such that D(x, E)= ‖x − e‖, then e is said to be a remotal, or farthest, point in E for x, and we define F (x, E)= {e ∈ E : D(x, E)= ‖x − e‖}. If F (x, E) = φ for all x ∈ X , then E is said to be a remotal set. The theory of remotal sets in Banach spaces is not as well as developed as that of proximinal sets; where the minimum distance is required to be attained. In [3], the authors proposed and discussed the following problem: Problem 1: When is a boundary point of E a remotal point? This seems to be a tough question and more general than Problem 2: When is an extreme point of E a remotal point? Recall that a point e ∈ E is said to be an extreme point of the convex set E, if e is not the middle point of any two other points of E. A special type of extreme points are exposed points. A point e ∈ E is said to be an exposed point of E, if there exists a linear functional f ∈ X ∗ , the dual space of the normed space X , such that f (y) <f (e) for all y ∈ E\{e}. Recall that, in this case, the set H := {x ∈ X : f (x)= f (e)} is called a supporting hyperplane of E at e; see [4]. In [1], it is proved that any normed linear space contains a bounded convex set whose exposed points are not necessarily remotal points. This is why we study here the problem: Problem 3: When is an exposed point of E a remotal point? We refer the reader to [3] and [1] for some results on this problem. The object of this paper is to address problem 3 above, where we give necessary and sufficient conditions for a class of exposed points to be remotal points in certain Banach spaces. In the sequel, X ∗ denotes the dual space of the normed space X , S (m, r) denotes the sphere centered at m with radius r and B(m, r) denotes the ball centered 2000 Mathematics Subject Classification. 46B20, 41A50, 41A65. Key words and phrases. Remotal sets, Approximation theory in Banach spaces. 1