On embeddings of finite metric spaces in $l_\infty^n$

TL;DR

This paper proves that large finite metric spaces can be isometrically embedded into lower-dimensional $l_infty$ spaces, reducing dimension by a fixed amount depending on the size of the space.

Contribution

It establishes a new dimension reduction result for finite metric spaces into $l_infty$, improving understanding of metric embeddings.

Findings

Any finite metric space on n points can be embedded into l_infty^{n-c} for sufficiently large n.

The embedding preserves distances exactly (isometric embedding).

The result holds for any fixed positive integer c.

Abstract

We prove that for any given integer $c>0$ any metric space on $n$ points may be isometrically embedded into $l_{\infty}^{n-c}$ provided $n$ is large enough.