Second Derivatives of a Convex Function and of Its Legendre-Fenchel Transformate

In 1977 Crouzeix established a simple relationship between the second-order differentials of a convex function $f:\mathbb{R}^n \to \mathbb{R} \cup \{ + \infty \}$ and its Legendre–Fenchel transformate $f^* :\mathbb{R}^n \to \mathbb{R} \cup \{ + \infty \}$. In the first part of this paper, the importance of Crouzeix’s formula is enhanced by illustrating how it can be applied to a large number of classical and modern mathematical problems. In the second part, the result of Crouzeix is extended to the case in which the functions f and $f^* $ are not necessarily smooth. This generalization is based on the works of Hiriart-Urruty and Seeger concerning the so-called second-order subdifferential of a convex function.