Hypomonotonicity of the Normal Cone and Proximal Smoothness

We study the properties of the normal cone to a proximally smooth set. We give a complete characterization of a proximally smooth set through the monotonicity properties of its normal cone in an arbitrary uniformly convex and uniformly smooth Banach space. We also give the exact bounds for the right-hand side in the monotonicity inequality for the normal cone in terms of the moduli of smoothness and convexity of a Banach space.