Global approximation of convex functions by differentiable convex functions on Banach spaces

2014/11/03 by Daniel Azagra, Azagra, Daniel, Carlos Mudarra +1
Computer Science · Mathematics · #26B25 #41A30 #46B20 #52A99 #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1411.0471

openalex publication_date 2014/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that if X is a Banach space whose dual X^* has an equivalent locally uniformly rotund (LUR) norm, then for every open convex U⊆ X, for every ε >0, and for every continuous and convex function f:U → ℝ (not necessarily bounded on bounded sets) there exists a convex function g:X → ℝ of class C¹(U) such that f-ε≤ g≤ f on U. We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) and convex functions by C^k smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by C^k smooth convex functions.

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