On the best Lipschitz extension problem for a discrete distance and the discrete infinity-Laplacian

Abstract. This paper is concerned with the best Lipschitz extension problem for a discrete distance that counts the number of steps. We relate this absolutely minimizing Lipschitz extension with a discrete ∞-Laplacian problem, which arise as the dynamic programming formula for the value function of some ε-tug-of-war games. As in the classical case, we obtain the absolutely minimizing Lipschitz extension of a datum f by taking the limit as p → ∞ in a nonlocal p– Laplacian problem. 1