We consider second-order objects for a convex function defined as the maximum of a finite number of C2-functions. Variational analysis yields explicit formulae for the second-order epi-derivatives of such max-functions. Another second-order object can be defined by means of a space decomposition that allows us to identify a subspace on which a Lagrangian related to a max-function is smooth. This decomposition yields an explicit expression for the so-called 𝒰-Hessian of the function, defined as the Hessian of the related Lagrangian. We show that the second-order epi-derivative relative to the 𝒰-subspace and the 𝒰-Hessian are equivalent second-order objects. Thus, the 𝒰-Lagrangian of a max-function captures the function's second-order epi-differential behavior with ordinary second derivatives.
