  We consider the problem of approximating the solution of variational problems
subject to the constraint that the admissible functions must be convex. This
problem is at the interface between convex analysis, convex optimization,
variational problems, and partial differential equation techniques.
  The approach is to approximate the (non-polyhedral) cone of convex functions
by a polyhedral cone which can be represented by linear inequalities. This
approach leads to an optimization problem with linear constraints which can be
computed efficiently, hundreds of times faster than existing methods.
