  This paper proves comparison principles for elliptic PDE involving the
Finsler infinity Laplacian, a second-order differential operator with
discontinuities in the gradient variable arising in $L^{\infty}$-variational
problems and tug-of-war games. The core of the paper consists in proving
generalized cone comparison principles. Among other consequences, these results
imply that, for any Finsler norm $\varphi$ in $\mathbb{R}^{d}$, a function $u$
is a $\varphi$-absolutely minimizing Lipschitz extension if and only if it is a
viscosity solution of the $\varphi$-infinity Laplace equation, settling a
longstanding question in the $L^{\infty}$-calculus of variations. The proofs
combine new geometric constructions with classical notions from convex
analysis.
