Coarse embeddability into Banach spaces

TL;DR

This paper surveys the field of coarse embeddability of metric spaces into Banach spaces, presents new results on non-embeddability into ll_2, and explores the complexity of embedding into ll_2.

Contribution

It provides a comprehensive survey and introduces new results on coarse embeddability, especially regarding non-embeddability into ll_2 and the difficulty of embedding into this space.

Findings

Non-embeddability into ll_2 may imply expander-like structures.

ll_2 is among the most challenging spaces to embed into.

New results on the limits of coarse embeddability into Banach spaces.

Abstract

The main purposes of this paper are (1) To survey the area of coarse embeddability of metric spaces into Banach spaces, and, in particular, coarse embeddability of different Banach spaces into each other; (2) To present new results on the problems: (a) Whether coarse non-embeddability into $\ell_2$ implies presence of expander-like structures? (b) To what extent $\ell_2$ is the most difficult space to embed into?