# Absolutely minimal semi–Lipschitz extensions

## TL;DR

This paper explores how to extend certain types of functions in asymmetric spaces, using two novel methods.

## Contribution

The paper introduces two new methods for constructing absolutely minimal semi-Lipschitz extensions in quasi-metric spaces.

## Key findings

- The Perron method can be adapted to the asymmetric case for optimal semi-Lipschitz extensions.
- An iteration scheme based on an unbalanced tug-of-war game provides a constructive approach for these extensions.
- The new method works even in symmetric metric spaces for Lipschitz functions.

## Abstract

The notion of quasi-metric space arises by revoking the symmetry from the definition of a distance. Semi-Lipschitz functions appear naturally as morphisms associated with the new structure. In this work, under suitable assumptions on the quasi-metric space (analogous to standard ones in the metric case), 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 setting 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 metric space, a constructive way of establishing existence of absolutely minimal Lipschitz extensions of real-valued Lipschitz functions.