A mean value formula for the variational $p$-Laplacian

We prove a new asymptotic mean value formula for the $p$-Laplace operator, $$\Delta_p u=\text{div}(|\nabla u|^{p-2}\nabla u),$$ valid in the viscosity sense. In the plane, and for a certain range of $p$, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.