Absolutely minimal semi–Lipschitz extensions

Abstract

The notion of quasi-metric space arises by revoking the symmetry from the definition of a distance.SemiLipschitz functions appear naturally as morphisms associated with the new structure. In this work we establish existence of optimal (that is,absolutely minimal) extensions of real valued semi Lipschitz functions from a subset of the space to the whole space. This is done in two different ways:first, by adapting the Perron method from the classical case to this asymmetric case and second, by means of an iteration scheme for (an unbalanced version of) the tug-of-war game, initiating the algorithm from a McShane extension. This new iteration scheme provides, even in the symmetric case of a metricspace, a constructive way of establishing existence of absolutely minimal Lipschitz extensions of real-valued Lipschitz functions.