On the equivalence of a Hessian-free inequality and Lipschitz continuous Hessian

2025/04/24 by Radu Ioan Boţ, Boţ, Radu I., Minh N. Dao +7
Mathematics · #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Optimization and Control (math.OC) #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.2504.17193

openalex publication_date 2025/04/24 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28

Abstract

It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuos function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fréchet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem.

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