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We investigate the connections between UC and UC* properties for orderedpairs of subsets (A,B) in metric spaces, which are involved in the study ofexistence 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 provethat these new notions are generalizations of the $UC$ property and that bothof them are sufficient for to ensure existence and uniqueness of best proximitypoints. We show that these two new notions are different from a uniformconvexity and even from a strict convexity. If we consider the underlying spaceto be a Banach space we find a sufficient condition which ensures that from theUC property it follows the uniform convexity of the underlying Banach space. Weillustrate the new notions with examples. We present an example of a cycliccontraction T in a space, which is not even strictly convex and the orderedpair (A,B) has not the UC property, but has the $BUC$ property and thus thereis a unique best proximity point of T in A.

On the UC and UC* properties and the existence of best proximity points in metric spaces