A study of farthest points and uniquely remotal sets

For a bounded subset K of a metric space (X, d), an element k(0) is an element of K is called a farthest point to an x is an element of X if d(x,k(0)) = sup(k is an element of K)d(x,k). The set of all farthest points to x in K is denoted by F-K(x). The set K is said to be remotal(uniquely remotal) if F-K(x) is non-empty(a singleton) for each x is an element of X. In this paper, we discuss the nature and structure of sets of farthest points and conditions under which sets are uniquely remotal. The underlying spaces are metric spaces and linear metric spaces.