Prox-regular functions have nonempty quadratic bundles, and generalized twice differentiability is equivalent to the classical twice differentiability of the associated Moreau envelope.
Attouch, Variational Convergence for Functions and Operators , Pitman, London, 1984
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Generalized Twice Differentiability and Quadratic Bundles in Second-Order Variational Analysis
Prox-regular functions have nonempty quadratic bundles, and generalized twice differentiability is equivalent to the classical twice differentiability of the associated Moreau envelope.