# Continuity of extensions of Lipschitz maps

## TL;DR

This paper investigates the conditions under which Lipschitz maps from subsets of Euclidean spaces can be extended to the whole space while preserving Lipschitz constants and distances, revealing key differences based on the target dimension.

## Contribution

It establishes the precise rate of continuity for Lipschitz extensions and characterizes when such extensions preserve uniform distances, highlighting differences between scalar and vector-valued cases.

## Key findings

- For $m>1$, Lipschitz maps are affine if such extensions exist.
- In the scalar case ($m=1$), any Lipschitz map admits such an extension.
- Extensions with preserved distances are possible when the difference between maps lies in a one-dimensional subspace and the set is geodesically convex.

## Abstract

We establish the sharp rate of continuity of extensions of $\mathbb{R}^m$-valued $1$-Lipschitz maps from a subset $A$ of $\mathbb{R}^n$ to a $1$-Lipschitz maps on $\mathbb{R}^n$. We consider several cases when there exists a $1$-Lipschitz extension with preserved uniform distance to a given $1$-Lipschitz map. We prove that if $m>1$ then a given map is $1$-Lipschitz and affine if and only if such distance preserving extension exists for any $1$-Lipschitz map defined on any subset of $\mathbb{R}^n$. This shows a striking difference from the case $m=1$, where any $1$-Lipschitz function has such property. Another example where we prove it is possible to find an extension with the same Lipschitz constant and the same uniform distance to another Lipschitz map $v$ is when the difference between the two maps takes values in a fixed one-dimensional subspace of $\mathbb{R}^m$ and the set $A$ is geodesically convex with respect to a Riemannian pseudo-metric associated with $v$.