  In this article we characterize the $\mathrm{L}^\infty$ eigenvalue problem
associated to the Rayleigh quotient $\left.{\|\nabla
u\|_{\mathrm{L}^\infty}}\middle/{\|u\|_\infty}\right.$ and relate it to a
divergence-form PDE, similarly to what is known for $\mathrm{L}^p$ eigenvalue
problems and the $p$-Laplacian for $p<\infty$. Contrary to existing methods,
which study $\mathrm{L}^\infty$-problems as limits of $\mathrm{L}^p$-problems
for $p\to\infty$, we develop a novel framework for analyzing the limiting
problem directly using convex analysis and geometric measure theory. For this,
we derive a novel fine characterization of the subdifferential of the
Lipschitz-constant-functional $u\mapsto\|\nabla u\|_{\mathrm{L}^\infty}$. We
show that the eigenvalue problem takes the form $\lambda \nu u
=-\operatorname{div}(\tau\nabla_\tau u)$, where $\nu$ and $\tau$ are
non-negative measures concentrated where $|u|$ respectively $|\nabla u|$ are
maximal, and $\nabla_\tau u$ is the tangential gradient of $u$ with respect to
$\tau$. Lastly, we investigate a dual Rayleigh quotient whose minimizers solve
an optimal transport problem associated to a generalized
Kantorovich--Rubinstein norm. Our results apply to all stationary points of the
Rayleigh quotient, including infinity ground states, infinity harmonic
potentials, distance functions, etc., and generalize known results in the
literature.
