Linear properties of Banach spaces and low distortion embeddings of metric graphs

TL;DR

This paper characterizes non-reflexive Banach spaces through low-distortion embeddings of metric graphs and explores conditions for $ ext{ell}_1^n$ to be nearly isomorphic to subspaces of Banach spaces.

Contribution

It provides a characterization of non-reflexive Banach spaces via metric graph embeddings and studies conditions for approximate isomorphism of $ ext{ell}_1^n$ in Banach spaces.

Findings

Non-reflexive Banach spaces characterized by low-distortion metric graph embeddings.

Conditions identified for $ ext{ell}_1^n$ to be nearly isomorphic to subspaces.

Analysis of non-linear conditions for approximate isomorphism.

Abstract

We characterize non-reflexive Banach spaces by a low-distortion (resp. isometric) embeddability of a certain metric graph up to a renorming. Also we study non-linear sufficient conditions for $\ell_1^n$ being $(1+\varepsilon)$-isomorphic to a subspace of a Banach space $X$.