Different forms of metric characterizations of classes of Banach spaces

TL;DR

This paper establishes an equivalence between embedding properties of Banach spaces and graph structures, providing a new metric characterization of classes of Banach spaces through finite graphs.

Contribution

It introduces a novel method linking Banach space embeddings with finite connected graphs, offering a new metric characterization framework.

Findings

Existence of finite connected graphs representing Banach space embeddings

Equivalence between uniform isomorphic and bilipschitz embeddings

Provides a new perspective on classifying Banach spaces

Abstract

For each sequence X of finite-dimensional Banach spaces there exists a sequence H of finite connected nweighted graphs with maximum degree 3 such that the following conditions on a Banach space Y are equivalent: (1) Y admits uniformly isomorphic embeddings of elements of the sequence X. (2) Y admits uniformly bilipschitz embeddings of elements of the sequence H.