On C⁰-fine approximation of convex functions by real analytic convex functions

2012/01/18 by Daniel Azagra, Azagra, Daniel
Computer Science · Mathematics · #26B25 #41A30 #46B20 #49N99 #52A1 #58E99 #Advanced Banach Space Theory #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1201.3858

openalex publication_date 2012/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that C⁰-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 C⁰-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 behavior. We give some applications concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

Related

• Global approximation of convex functions
• Inner and outer smooth approximation of convex hypersurfaces. When is it possible?
• Global approximation of convex functions by differentiable convex functions on Banach spaces
• Smooth maps on convex sets
• Compensated Convexity Methods for Approximations and Interpolations of Sampled Functions in Euclidean Spaces: Theoretical Foundations