Whitney extension theorems for convex functions of the classes C1 and C1,ω

Abstract

Let C be a subset of R n (not necessarily convex), f : C → R be a function and G : C → R n be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f , G for the existence of a convex function F ∈ C 1,ω (R n ) such that F = f on C and ∇F = G on C, with a good control of the modulus of continuity of ∇F in terms of that of G. On the other hand, assuming that C is compact, we also solve a similar problem for the class of C 1 convex functions on R n , with a good control of the Lipschitz constants of the extensions (namely, Lip(F ) G ∞). Finally, we give a geometrical application concerning interpolation of compact subsets K of R n by boundaries of C 1 or C 1,1 convex bodies with prescribed outer normals on K.

On the global shape of continuous convex functions on Banach spaces

“…Thus one could say that, up to an additive linear perturbation and a composition with a linear projection onto a subspace of possibly smaller dimension, every convex function on R n is essentially coercive. This decomposition property has been useful in the proofs of several recent results on global smooth approximation and extension by convex functions; see [1,4,5,2,3].…”

Extensions of convex functions with prescribed subdifferentials

“…Very recently, a related problem has been solved for convex functions of the classes C 1 (R n ) and C 1,ω (X) (for a Hilbert space X) in the situation where the mapping G is single-valued and one additionally requires that the extension F be of class C 1 (R n ) (which amounts to asking that ∂F (x) be a singleton for every x ∈ R n ) or of class C 1,ω (X); see [1,2,3]. A solution to a similar problem for general (not necessarily convex) functions was given in [11,Theorem 5], characterizing the pairs f : E → R, G : E ⇒ R n with f continuous and G upper semicontinuous and nonempty, compact and convexvalued which admit a (generally nonconvex) extension F of f whose Fréchet subdifferential is upper semicontinuous on R n and extends G from E.…”

Global geometry and C1 convex extensions of 1-jets

“…Given C a differentiability class in R n , E a subset of R n , and functions f : E → R and G : E → R n , how can we decide whether there is a convex function F ∈ C such that F (x) = f (x) and ∇F (x) = G(x) for all x ∈ E? This is a natural question which we could solve in [5] in the case that C = C 1,ω (R n ), where ω : [0, ∞) → [0, ∞) is a (strictly increasing and concave) modulus of continuity. A necessary and sufficient condition is that there exists a constant M > 0 such that…”

“…Let C 1 conv (R n ) stand for the set of all functions f : R n → R which are convex and of class C 1 . In [5], and for the class C = C 1 (R n ), we could only obtain a solution to Problem 1.1 in the particular case that E is a compact set. In this special situation the three necessary and sufficient conditions on (f, G) that we obtained for C 1 conv (R n ) extendibility are: (which tells us that if two points of the graph of f lie on a line segment contained in a hyperplane which we want to be tangent to the graph of an extension at one of the points, then our putative tangent hyperplanes at both points must be the same).…”