Subdifferential representation of convex functions on X^*

2017/11/28 by Duanxu Dai, Dai, Duanxu
Mathematics · #26E25 #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Inequalities and Applications #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Primary 52A41 #Secondary 46B20

paper · pdf · doi:10.48550/arxiv.1711.10161

openalex publication_date 2017/11/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we obtain subdifferential representation of a proper w^*-lower semicontinous convex function on X^* as follows: Let g be a proper convex w^*-lower semicontinuous function on X^*. Assume that int dom g ≠∅ (resp. int (dom (g^*|_X))≠∅). Then given any point x₀^* ∈ D (∂ g∩ X) and x^* ∈ dom g (resp. x^*∈ X^*), we have g(x^*)=g(x₀^*)+sup\∑_i=0^n-1⟨ x_i,x_i+1^*-x_i^*⟩ +⟨ x_n,x^*-x_n^*⟩ \, where the above supremum is taken over all integers n, all x_i^*∈ X^* and all x_i∈∂ g(x_i^*)∩ X for i=0,1,⋯,n. (resp. if, moreover, X^* has the Radon-Nikodym property, then we may estimate the above supremum among the set of w^*-strongly exposed points of g.)

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