Impossibility of almost extension

TL;DR

This paper establishes the optimal bounds for almost extension of Lipschitz functions between normed spaces, showing that Bourgain's theorem cannot be significantly improved and identifying sharp bounds in Euclidean spaces.

Contribution

The paper proves that Bourgain's almost extension bounds are essentially tight, providing matching lower bounds and extending results specifically for Euclidean spaces.

Findings

Bourgain's upper bound is optimal up to lower order factors.

In Euclidean spaces, the approximation can be improved to rom or general spaces.

Sharp lower bounds are established for both general normed spaces and Euclidean spaces.

Abstract

Let $({\mathbf X},\|\cdot\|_{\mathbf X}), ({\mathbf Y},\|\cdot\|_{\mathbf Y})$ be normed spaces with ${\mathrm{dim}}({\mathbf X})=n$. Bourgain's almost extension theorem asserts that for any ${\varepsilon}>0$, if ${\mathcal{N}}$ is an ${\varepsilon}$-net of the unit sphere of ${\mathbf X}$ and $f:{\mathcal{N}}\to {\mathbf Y}$ is $1$-Lipschitz, then there exists an $O(1)$-Lipschitz $F:{\mathbf X}\to {\mathbf Y}$ such that $\|F(a)-f(a)\|_{\mathbf Y}\lesssim n{\varepsilon}$ for all $a\in \mathcal{N}$. We prove that this is optimal up to lower order factors, i.e., sometimes $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim n^{1-o(1)}{\varepsilon}$ for every $O(1)$-Lipschitz $F:{\mathbf X}\to {\mathbf Y}$. This improves Bourgain's lower bound of $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim n^{c}{\varepsilon}$ for some $0<c<\frac12$. If ${\mathbf X}=\ell_2^n$, then the approximation in the almost extension theorem can be improved to $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\lesssim \sqrt{n}{\varepsilon}$. We prove that this is sharp, i.e., sometimes $\max_{a\in {\mathcal{N}}} \|F(a)-f(a)\|_{\mathbf Y}\gtrsim \sqrt{n}{\varepsilon}$ for every $O(1)$-Lipschitz $F:\ell_2^n\to {\mathbf Y}$.