  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.
