A locally strongly convex function with Lipschitz derivative exists on a Banach space only if the space is isomorphic to a Hilbert space.
Legendre Forms in Reflexive Banach Spaces
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Characterization of Hilbertizable spaces via convex functions
A locally strongly convex function with Lipschitz derivative exists on a Banach space only if the space is isomorphic to a Hilbert space.