No dimension reduction for doubling subsets of $\ell_q$ when $q>2$ revisited

TL;DR

This paper revisits and simplifies the proof of the impossibility of dimension reduction for doubling subsets of bb_q when q>2, and extends the results to other geometric spaces.

Contribution

It provides an elementary, more general proof of dimension reduction impossibility for bb_q spaces with q>2, and extends to non-positively curved and uniformly convex Banach spaces.

Findings

Elementary proof of dimension reduction impossibility for bb_q, q>2

Extension of obstructions to non-positively curved spaces

Extension of obstructions to asymptotically uniformly convex Banach spaces

Abstract

We revisit the main results from \cites{BGN_SoCG14,BGN_SIAM15} and \cite{LafforgueNaor14_GD} about the impossibility of dimension reduction for doubling subsets of $\ell_q$ for $q>2$. We provide an alternative elementary proof of this impossibility result that combines the simplicity of the construction in \cites{BGN_SoCG14,BGN_SIAM15} with the generality of the approach in \cite{LafforgueNaor14_GD} (except for $L_1$ targets). One advantage of this different approach is that it can be naturally generalized to obtain embeddability obstructions into non-positively curved spaces or asymptotically uniformly convex Banach spaces.