Bi-Lipschitz approximation by finite-dimensional imbeddings

TL;DR

This paper demonstrates that Riemannian manifolds can be approximated by finite-dimensional bi-Lipschitz embeddings, refining Gromov's systolic inequality proof using non-standard analysis techniques.

Contribution

It introduces a method to approximate Kuratowski embeddings with finite-dimensional bi-Lipschitz maps for any desired accuracy, utilizing first variation formulas in non-standard analysis.

Findings

Finite-dimensional bi-Lipschitz approximations exist for Kuratowski embeddings.

The approximation can be made arbitrarily close with (1+C)-bi-Lipschitz maps.

The approach leverages non-standard analysis and first variation formulas.

Abstract

We show that the Kuratowski imbedding of a Riemannian manifold in L^\infty, exploited in Gromov's proof of the systolic inequality for essential manifolds, admits an approximation by a (1+C)-bi-Lipschitz (onto its image), finite-dimensional imbedding for every C>0. Our key tool is the first variation formula thought of as a real statement in first-order logic, in the context of non-standard analysis.