Variational inequalities and regularity properties of closed sets in Hilbert spaces

Some properties of closed sets which generalize concepts of Convex Analysis are compared and chararterized. Some of them have a global character and a re concerned with controlling the lack of monotonicity of the Frechet subdi fferential of the indicator function. The connection with the local structu re of sets in finite as well as in infinite dimensional spaces is also inve stigated. Special emphasis is given to a class of sets satisfying an extern al sphere condition, with locally uniform radius.