A general theorem of existence of quasi absolutely minimal Lipschitz extensions

TL;DR

This paper establishes a general theorem ensuring the existence of quasi absolutely minimal Lipschitz extensions for a broad class of Lipschitz extension problems, under mild conditions, expanding the theoretical understanding of such extensions.

Contribution

It proves that if a minimal Lipschitz extension exists, then a quasi absolutely minimal Lipschitz extension also exists under mild conditions, generalizing previous results.

Findings

Existence of quasi absolutely minimal Lipschitz extensions under mild conditions

Extension applies to a wide class of generalized Lipschitz problems

Nearly minimal extensions can be constructed with arbitrarily small deviation

Abstract

In this paper we consider a wide class of generalized Lipschitz extension problems and the corresponding problem of finding absolutely minimal Lipschitz extensions. We prove that if a minimal Lipschitz extension exists, then under certain other mild conditions, a quasi absolutely minimal Lipschitz extension must exist as well. Here we use the qualifier "quasi" to indicate that the extending function in question nearly satisfies the conditions of being an absolutely minimal Lipschitz extension, up to several factors that can be made arbitrarily small.