Daniel Azagra

PERTURBED SMOOTH LIPSCHITZ EXTENSIONS OF UNIFORMLY CONTINUOUS FUNCTIONS ON BANACH SPACES

We show that if Y is a separable subspace of a Banach space X such that both X and the quotient X/Y have C-p-smooth Lipschitz bump functions, and U is a bounded open subset of X, then, for every uniformly continuous function f : Y boolean AND U --> R and every epsilon > 0, there exists a C-p-smooth Lipschitz function F : X --> R such that |F(y)- f( y)| less than or equal to epsilon for every y is an element of Y boolean AND U.

Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds

We prove that every continuous mapping from a separable infinite-dimensional Hilbert space X into R-m can be uniformly approximated by C-infinity-smooth mappings with no critical points. This kind of result can be regarded as a sort of strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold that is a level set of a smooth function with no critical points. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are C-infinity-smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modeled on X.

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.

On the range of the derivatives of a smooth function between Banach spaces

We study the size of the range of the derivatives of a smooth function between Banach spaces. We establish conditions on a pair of Banach spacesX andY to ensure the existence of a C p smooth (Fr echet smooth or a continuous G^ ateaux smooth) function f from X onto Y such that f vanishes outside a bounded set and all the derivatives off are surjections. In particular we deduce the following results. For the G^ ateaux case, whenX andY are separable andX is innite-dimensional, there exists a continuous G^ ateaux smooth functionf fromX toY , with bounded support, so that f 0 (X )= L(X;Y ). In the Fr echet case, we get that if a Banach spaceX has aF r echet smooth bump and densX = dens L(X;Y ), then there is a Fr echet smooth function f:X! Y with bounded support so that f 0 (X )= L(X;Y ). Moreover, we see that if X has a C p smooth bump with bounded derivatives and densX = dens L m (X;Y ) then there exists another C p smooth function f:X! Y so that f (k) (X )= L k(X;Y ) for all k =0 ; 1;:::;m. As an application, we show that every bounded starlike body on a separable Banach space X with a (Fr echet or G^ ateaux) smooth bump can be uniformly approximated by smooth bounded starlike bodies whose cones of tangent hyperplanes ll the dual space X. In the non-separable case, we prove that X has such property if X has smooth partitions of unity.

Smooth Lipschitz Retractions of Starlike Bodies onto their Boundaries in Infinite-Dimensional Banach Spaces

Let X be an infinite-dimensional Banach space, and let A be a CP Lipschitz bounded starlike body (for instance the unit ball of a smooth norm). We prove that.

On the topological classification of starlike bodies in Banach spaces

Abstract Starlike bodies are interesting in nonlinear analysis because they are strongly related to polynomials and smooth bump functions, and their topological and geometrical properties are therefore worth studying. In this note we consider the question as to what extent the known results on topological classification of convex bodies can be generalized for the class of starlike bodies, and we obtain two main results in this line, one which follows the traditional Bessaga–Klee scheme for the classification of convex bodies (and which in this new setting happens to be valid only for starlike bodies whose characteristic cones are convex), and another one which uses a new classification scheme in terms of the homotopy type of the boundaries of the starlike bodies (and which holds in full generality provided the Banach space is infinite-dimensional).

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).

Global geometry and C1 convex extensions of1-jets

Let

Subdifferentiable functions satisfy Lusin properties of class C1 or C2

E

Nonsmooth analysis and Hamilton–Jacobi equations on Riemannian manifolds

be an arbitrary subset of