The Dirichlet problem for the convex envelope

The Convex Envelope of a given function was recently characterized as the solution of a fully nonlinear Partial Differential Equation (PDE). In this article we study a modified problem: the Dirichlet problem for the underlying PDE. The main result is an optimal regularity result. Differentiability ($C^{1,\alpha}$ regularity) of the boundary data implies the corresponding result for the solution in the interior, despite the fact that the solution need not be continuous up to the boundary. Secondary results are the characterization of the convex envelope as: (i) the value function of a stochastic control problem, and (ii) the optimal underestimator for a class of nonlinear elliptic PDEs.