Global Lipschitz extension preserving the slope

TL;DR

This paper proves that Lipschitz functions can be extended from subsets to entire metric spaces while approximately preserving their slope and Lipschitz constant, addressing a question in metric analysis.

Contribution

It introduces a method to extend Lipschitz functions preserving slopes and Lipschitz constants, solving an open problem in metric space analysis.

Findings

Lipschitz functions can be extended while preserving slopes.

Extension preserves the Lipschitz constant up to a small error.

Results apply to ascending and descending slopes.

Abstract

We show that every real-valued Lipschitz function on a subset of a metric space can be extended to the whole space while preserving the slope and, up to a small error, the global Lipschitz constant. This answers a question posed by Di Marino, Gigli, and Pratelli, who established the analogous property for the asymptotic Lipschitz constant. We also prove the same result for the ascending slope and for the descending slope.