Global and fine approximation of convex functions

Abstract

Abstract. Let U ⊆ Rd be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f : U → R can be approximated by real analytic convex functions, uniformly on all of U . We also show that C 0 -fine approximation of convex functions by smooth (or real analytic) convex functions on R d is possible in general if and only if d = 1. Nevertheless, for d ≥ 2 we give a characterization of the class of convex functions on R d which can be approximated by real analytic (or just smoother) convex functions in the C 0 -fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every C 1 convex and proper function on U can be approximated by C ∞ convex functions in the C 1 -fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

On the global shape of continuous convex functions on Banach spaces

“…A strongly related kind of nice convex functions is what one can call essentially coercive convex functions, namely convex functions which are coercive up to linear perturbation. Of course, even in the case Z = R n , not every convex function is essentially coercive, but it is nonetheless true (see [1,Lemma 4.2] and [5,Theorem 1.11]) that every convex function f : R n → R admits a decomposition of the form…”

“…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].…”

On a Hilbert space of entire functions

“…Propositions 4 and 5 can be strengthen by employing the results by D. Azagra [2], [3]. He proved the following theorem.…”

A minimax problem for sums of translates on the torus

“…We need another lemma, similar to [, Theorem 1], in order to be able to apply the previous result. Lemma Let be a strictly concave, non‐increasing function.…”

“…, the maximum of F (x, ·) on I 2 (x) = [x 2 , x 1 ] is attained at z 2 (x) = π and m 2 (x) = F (x, π) = π + επ 2 (6 √ 2 − 7), the maximum of F (x, ·) on I 3…”