The paper presents results on viscosity solutions of boundary value problems for the so-called infinity Laplacian PDE. Among the results are the characterization of the almost minimizing Lipschitz extension property by means of solutions of such equations as well as several regularity properties for the solutions. While some of the results are already known in the literature [see, e.g., \textit{R. Jensen}, ``Uniqueness of Lipschitz extensions: Minimizing the sup norm of the gradient'', Arch. Ration. Mech. Anal. 123, No. 1, 51-74 (1993; Zbl 0789.35008)], the main contribution of the paper is a new proof technique working directly with the infinity Laplacian and a ``comparison with cones'' property, thus avoiding to take limits of \(p\)-Laplacians.