Non-Convergence Result for Conformal Approximation of Variational Problems Subject to a Convexity Constraint

In this article, we are interested in the minimization of functionals in the set of convex functions. We investigate the discretization of the convexity through various numerical methods and find a geometrical obstruction confirmed by numerical simulations. We prove that there exist some convex functions that cannot be the limit of any conformal P 1 Finite Element sequence for a wide variety of refined meshes.