  We investigate the connections between UC and UC* properties for ordered
pairs of subsets (A,B) in metric spaces, which are involved in the study of
existence and uniqueness of best proximity points. We show that the $UC^{*}$
property is included into the UC property. We introduce some new notions:
bounded UC (BUC) property and uniformly convex set about a function. We prove
that these new notions are generalizations of the $UC$ property and that both
of them are sufficient for to ensure existence and uniqueness of best proximity
points. We show that these two new notions are different from a uniform
convexity and even from a strict convexity. If we consider the underlying space
to be a Banach space we find a sufficient condition which ensures that from the
UC property it follows the uniform convexity of the underlying Banach space. We
illustrate the new notions with examples. We present an example of a cyclic
contraction T in a space, which is not even strictly convex and the ordered
pair (A,B) has not the UC property, but has the $BUC$ property and thus there
is a unique best proximity point of T in A.
