Characterization of Hilbertizable spaces via convex functions

TL;DR

This paper establishes that certain convex functions with smoothness properties on Banach spaces imply the space is isomorphic to a Hilbert space, providing new characterizations of Hilbertizable spaces.

Contribution

It introduces novel conditions involving convex functions and their conjugates that characterize Hilbert spaces among Banach spaces.

Findings

Existence of a strongly convex function with Lipschitz derivative implies Hilbert space structure.

Both a convex function and its conjugate being $C^2$ imply the space is Hilbertizable.

Provides new convex-analytic criteria for identifying Hilbert spaces.

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 $C^2$ then the underlying space is also isomorphic to a Hilbert space.