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Absolutely minimal semi-Lipschitz extensions.

Aris Daniilidis1, Trí Minh Lê1, Francisco M Venegas2

  • 1Institut für Stochastik und Wirtschaftsmathematik, VADOR E105-04 TU Wien, Wiedner Hauptstraße 8, A-1040 Wien, Austria.

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|October 20, 2025
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Summary

This study establishes optimal extensions for semi-Lipschitz functions in quasi-metric spaces. Two novel methods, including an unbalanced tug-of-war game, prove the existence of these essential mathematical extensions.

Keywords:
Primary 26A16, 39B82Secondary 35B50, 41A05

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Area of Science:

  • Mathematics
  • Analysis
  • Topology

Background:

  • Quasi-metric spaces generalize metric spaces by relaxing the symmetry property of distance.
  • Semi-Lipschitz functions are natural mappings in these asymmetric spaces.
  • Extending functions while preserving properties is a fundamental problem in analysis.

Purpose of the Study:

  • To establish the existence of optimal (absolutely minimal) extensions of real-valued semi-Lipschitz functions.
  • To adapt existing methods and introduce new ones for function extension in quasi-metric settings.
  • To provide constructive proofs for the existence of minimal extensions.

Main Methods:

  • Adaptation of the Perron method to the asymmetric quasi-metric context.
  • Development of an iterative scheme based on an unbalanced tug-of-war game.
  • Utilizing McShane extensions as starting points for the iterative scheme.

Main Results:

  • Existence of absolutely minimal extensions for semi-Lipschitz functions in quasi-metric spaces is proven.
  • The Perron method adaptation successfully yields these extensions.
  • The novel tug-of-war iteration scheme provides a constructive existence proof, applicable even to metric spaces.

Conclusions:

  • The study successfully extends the theory of function extensions to quasi-metric spaces.
  • New analytical tools (Perron method adaptation, unbalanced tug-of-war) are introduced.
  • The findings offer constructive methods for obtaining minimal Lipschitz extensions in both quasi-metric and metric spaces.