Convex C¹ extensions of 1-jets from compact subsets of Hilbert\n spaces

2019/11/11 by Daniel Azagra, Azagra, Daniel, Carlos Mudarra +1
Computer Science · Mathematics · #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Functional Equations Stability Results #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.1911.04166

openalex publication_date 2019/11/11 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28

Abstract

Let X denote a Hilbert space. Given a compact subset K of X and two\ncontinuous functions f:K\→\ℝ, G:K\→ X, we show that a necessary\nand sufficient condition for the existence of a convex function F\∈ C¹(X)\nsuch that F=f on K and \∇ F=G on K is that the 1-jet (f, G)\nsatisfies (1) f(x)\≥ f(y)+ \⟨ G(y), x-y\⟩ for all x, y\∈ K,\nand (2) if x, y\∈ K and f(x)= f(y)+ \⟨ G(y), x-y\⟩ then\nG(x)=G(y). We also solve a similar problem for K replaced with an arbitrary\nbounded subset of X, and for C¹(X) replaced with the class\nC^1,u_b(X) of differentiable functions with uniformly continuous\nderivatives on bounded subsets of X.\n

Related

• Locally C^1,1 convex extensions of 1-jets
• Explicit formulas for C^{1, 1} and C^{1, ω}_\textrmconv extensions of 1-jets in Hilbert and superreflexive spaces
• C^1,ω extension formulas for 1-jets on Hilbert spaces
• An Extension Theorem for convex functions of class C^1,1 on Hilbert\n spaces
• Differentiable convex extensions with sharp Lipschitz constants