Finite determination for embeddings into Banach spaces: new proof, low distortion

TL;DR

This paper improves the finite determination results for embeddings into Banach spaces by significantly reducing distortion from about 3000 to just over 3 and simplifies the proof process using new techniques.

Contribution

It introduces a new logarithmic spiral gluing method that drastically lowers distortion and simplifies the proof of finite determination for embeddings into Banach spaces.

Findings

Distortion reduced from ~3000 to 3+ε

Simplified proof using Brunel-Sucheston result

Enhanced methods for embedding finite metric spaces

Abstract

The main goal of this paper is to improve the result of Ostrovskii (2012) on the finite determination of bilipschitz and coarse embeddability of locally finite metric spaces into Banach spaces. There are two directions of the improvement: (1) Substantial decrease of distortion (from about $3000$ to $3+\ep$) is achieved by replacing the barycentric gluing by the logarithmic spiral gluing. This decrease in the distortion is particularly important when the finite determination is applied to construction of embeddings. (2) Simplification of the proof: a collection of tricks employed in Ostrovskii (2012) is no longer needed, in addition to the logarithmic spiral gluing we use only the Brunel-Sucheston result on existence of spreading models.