Juan Ferrera

Every closed convex set is the set of minimizers of some ^{∞}-smooth convex function

We show that for every closed convex set C in a separable Banach space X there is a C∞-smooth convex function f: X → [0,∞) so that f -1 (0) = C. We also deduce some interesting consequences concerning smooth approximation of closed convex sets and continuous convex functions.

Approximate Rolle's theorems for the proximal subgradient and the generalized gradient

We establish approximate Rolles theorems for the proximal subgradient and for the generalized gradient. We also show that an exact Rolles theorem for the generalized gradient is completely false in all infinite-dimensional Banach spaces (even when they do not possess smooth bump functions).

Subdifferentiable functions satisfy Lusin properties of class C1 or C2

Abstract Let f : R n → R be a function. Assume that for a measurable set Ω and almost every x ∈ Ω there exists a vector ξ x ∈ R n such that lim inf h → 0 f ( x + h ) − f ( x ) − 〈 ξ x , h 〉 | h | 2 > − ∞ . Then we show that f satisfies a Lusin-type property of order 2 in Ω , that is to say, for every e > 0 there exists a function g ∈ C 2 ( R n ) such that L n ( { x ∈ Ω : f ( x ) ≠ g ( x ) } ) ≤ e . In particular every function which has a nonempty proximal subdifferential almost everywhere also has the Lusin property of class C 2 . We also obtain a similar result (replacing C 2 with C 1 ) for the Frechet subdifferential. Finally we provide some examples showing that these kinds of results are no longer true for Taylor subexpansions of higher order.