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

This paper concerns 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 infinity-Laplacian problem, which arises as the dynamic programming formula for the value function of some c-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 -> infinity in a nonlocal p-Laplacian problem. (C) 2011 Elsevier Masson SAS. All rights reserved.