Whitney Extension Theorems for convex functions of the classes C¹ and C^1,ω

2015/07/14 by Daniel Azagra, Azagra, Daniel, Carlos Mudarra +1 · 3 citations
Computer Science · Mathematics · #26B05 #35J96 #52A20 #52A41 #53A99 #53C45 #54C20 #58C25 #Advanced Banach Space Theory #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1507.03931

openalex publication_date 2015/07/14 · openalex created_date 2022/09/01 · openalex updated_date 2026/07/28

Abstract

Let C be a subset of ℝⁿ (not necessarily convex), f:C→ℝ be a function, and G:C→ℝⁿ 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}(ℝⁿ) 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 C¹ convex functions on ℝⁿ, with a good control of the Lipschitz constants of the extensions (namely, \textrmLip(F)\lesssim ‖G‖_∞). Finally, we give a geometrical application concerning interpolation of compact subsets K of ℝⁿ by boundaries of C¹ or C^1,1 convex bodies with prescribed outer normals on K.

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