Linear Lipschitz and $C^1$ extension operators through random projection

TL;DR

This paper introduces a method using random projections to extend Lipschitz and $C^1$ functions from metric spaces to Banach spaces, generalizing classical extension theorems.

Contribution

It constructs a regular random projection technique for metric space extensions, providing a more direct proof and broadening the scope of Whitney's $C^1$ extension theorem.

Findings

Constructed a regular random projection for metric space extension

Extended Lipschitz and $C^1$ functions linearly

Generalized Whitney's $C^1$ extension theorem to Banach spaces

Abstract

We construct a regular random projection of a metric space onto a closed doubling subset and use it to linearly extend Lipschitz and $C^1$ functions. This way we prove more directly a result by Lee and Naor and we generalize the $C^1$ extension theorem by Whitney to Banach spaces.