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Journal of Convex Analysis 30 (2023), No. 4, 1319--1328
Copyright Heldermann Verlag 2023

A Simple Proof of the Baillon-Haddad Theorem on Open Subsets of Hilbert Spaces

Daniel Wachsmuth
Institut für Mathematik, Universität Würzburg, Germany

Gerd Wachsmuth
Institut für Mathematik, Brandenburgische Technische Universität, Cottbus-Senftenberg, Germany

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.

Keywords: Baillon-Haddad theorem, cocoercivity, strong smoothness.

MSC: 26B25, 47H05, 47N10, 49J50.

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