Lusin-type properties of convex functions and convex bodies

2020/11/20 by Azagra, Daniel, Hajłasz, Piotr · 1 citation
#26B25 #28A75 #41A30 #52A20 #52A27 #53C45 #Classical Analysis and ODEs (math.CA) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2011.10279

Abstract

We prove that if f:ℝⁿ→ℝ is convex and A⊂ℝⁿ has finite measure, then for any ε>0 there is a convex function g:ℝⁿ→ℝ of class C^1,1 such that Lⁿ(\x∈ A: f(x)≠ g(x)\)0, there exists a convex body W_ε of class C^1,1 such that H^n-1(∂ W∖ ∂ W_ε)< ε. We also show that if f:ℝⁿ→ℝ is a convex function and f is not of class C^1,1_{\rm loc}, then for any ε>0 there is a convex function g:ℝⁿ→ℝ of class C^1,1_{\rm loc} such that Lⁿ(\x∈ ℝⁿ: f(x)≠ g(x)\)0 there exists a convex hypersurface S_ε of class C^1,1_\textrmloc such that H^n-1(S∖ S_ε)

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• C²-Lusin approximation of convex functions: one variable case

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