I will characterize the L∞ eigenvalue problem which is solved by stationary points of the Rayleigh quotient ∥∇u∥L∞/∥u∥∞ and relate it to a divergence-form PDE, similarly to what is known for Lp eigenvalue problems and the p-Laplacian for p < ∞. Contrary to most existing methods, which study L∞-problems as limits of Lp-problems for large values of p, I shall present a novel framework for analyzing the limiting problem directly using convex analysis and measure theory. Our results rely on a novel fine characterization of the subdifferential of the Lipschitz-constant-functional. I also study a dual Rayleigh quotient whose minimizers solve an optimal transport problem associated to a generalized Kantorovich--Rubinstein norm. This is joint work with Yury Korolev and based on the article (https://arxiv.org/abs/2107.12117).