Classification of finite metric spaces and combinatorics of convex polytopes

TL;DR

This paper establishes a canonical link between finite metric spaces and symmetric convex polytopes, framing metric space classification through the combinatorial properties of these polytopes.

Contribution

It introduces a novel correspondence between finite metric spaces and convex polytopes, providing a new approach to classify metric spaces via polytope combinatorics.

Findings

Established a canonical correspondence between finite metric spaces and symmetric convex polytopes

Formulated the classification problem of metric spaces in terms of polytope combinatorics

Provided a framework connecting metric space theory and convex geometry

Abstract

We describe the canonical correspondence between set of all finite metric spaces and set of special symmetric convex polytopes, and formulate the problem about classification of the metric spaces in terms of combinatorial structure of those polytopes.