Characterization of Hilbertizable spaces via convex functions

2025/06/05 by Nicolas Borchard, Gerd Wachsmuth, Borchard, Nicolas +1
Mathematics · Computer Science · #Advanced Banach Space Theory #Optimization and Variational Analysis #Functional Equations Stability Results

paper · pdf · doi:10.48550/arxiv.2506.04686

Abstract

We show that the existence of a strongly convex function with a Lipschitz derivative on a Banach space already implies that the space is isomorphic to a Hilbert space. Similarly, if both a function and its convex conjugate are C² then the underlying space is also isomorphic to a Hilbert space.

Related

• On some new metric characterisations of Hilbert spaces
• A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces
• Hilbert domains quasi-isometric to normed vector spaces
• Some Banach spaces are almost Hilbert
• Convex-transitivity and function spaces