A parametric smooth variational principle and support properties of convex sets and functions

We show a modified version of Georgievʹs parametric smooth variational principle, and we use it to derive new support properties of convex functions and sets. For example, our results imply that, for any proper l.s.c. convex nonaffine function h on a Banach space Y, D ( ∂ h ) is pathwise connected and R ( ∂ h ) has cardinality at least continuum. If, in addition, Y is Fréchet-smooth renormable, then R ( ∂ h ) is pathwise connected and locally pathwise connected. Analogous properties for support points and normalized support functionals of closed convex sets are proved; they extend and strengthen recent results proved by C. De Bernardi and the author for bounded closed convex sets.