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A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces

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arxiv 2204.00282 v1 pith:A4VZNH3Q submitted 2022-04-01 math.FA math.OC

classification math.FAmath.OC
keywords convexopenspacessubsetsbaillon-haddadfunctionshilbertproof
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We give a simple proof of the Baillon-Haddad theorem for convex functions defined on open and convex subsets of Hilbert spaces. We also state some generalizations and limitations. In particular, we discuss equivalent characterizations of the Lipschitz continuity of the derivative of convex functions on open and convex subsets of Banach spaces.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

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

    math.OC 2025-04 accept novelty 7.0 of 10

    If a continuous map between a Hilbert and a reflexive Banach space satisfies the Hessian-free Jensen inequality, then it is Frechet differentiable and its derivative is Lipschitz continuous.

  2. Mirror Descent on Reproducing Kernel Banach Spaces

    cs.LG 2024-11 conditional novelty 5.0 of 10

    The authors design a functional mirror descent for reproducing kernel Banach spaces and prove conditional linear and O(1/√t) convergence, with a finite-center p-norm RKBS instantiation.

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