We analyse the functional 𝒥 (u)=∥∇ u∥∞ defined on Lipschitz functions with homogeneous Dirichlet boundary conditions. Our analysis is performed directly on the functional without the need to approximate with smooth p-norms. We prove that its ground states coincide with multiples of the distance function to the boundary of the domain. Furthermore, we compute the L 2-subdifferential of 𝒥 and characterize the distance function as the unique nonnegative eigenfunction of the subdifferential operator. We also study properties of general eigenfunctions, in particular their nodal sets. Furthermore, we prove that the distance function can be computed as the asymptotic profile of the gradient flow of 𝒥 and construct analytic solutions of fast marching type. In addition, we give a geometric characterization of the extreme points of the unit ball of 𝒥.