We extend the principle of comparison with cones introduced by Crandall, Evans and Gariepy in  for the Minimizing Lipschitz Extension Problem to a wide class of supremal functionals. This gives a geometrical characterization of the absolute minimizers (optimal solutions whose minimality is local). Some application to the stability of absolute minimizers with respect to the Gamma-convergence is given. A variation of the basic idea also allows to characterize the minimal Lipschitz extensions in length metric spaces.
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