Optimal Lipschitz extensions and the infinity laplacian

We reconsider in this paper boundary value problems for the so-called "infinity Laplacian" PDE and the relationships with optimal Lipschitz extensions of the boundary data. We provide some fairly elegant new proofs, which clarify and simplify previous work, and in particular draw attention to the fact that solutions may be characterized by a comparison principle with appropriate cones. We in particular show how comparison with cones directly implies the variational principle associated with the equation. In addition, we establish a Liouville theorem for subsolutions bounded above by planes.