Combinatorics of the Lipschitz polytope

TL;DR

This paper studies the combinatorial structure of Lipschitz polytopes associated with metrics on finite sets, revealing explicit face counts for generic metrics and linking these to root polytope triangulations, with bounds on metric classification complexity.

Contribution

It provides exact formulas for the face counts of Lipschitz polytopes for generic metrics and establishes bounds on the number of metric classes based on combinatorial complexity.

Findings

Number of (n-m)-dimensional faces equals (n+m)!/m!m!(n-m)! for generic metrics.

Linked face counts to regular triangulations of the root polytope.

Established bounds on the logarithm of the number of metric classes as n^3 log n and n^2.

Abstract

Let $\rho$ be a metric on the set $X=\{1,2,\dots,n+1\}$. Consider the $n$-dimensional polytope of functions $f:X\rightarrow \mathbb{R}$, which satisfy the conditions $f(n+1)=0$, $|f(x)-f(y)|\leq \rho(x,y)$. The question on classifying metrics depending on the combinatorics of this polytope have been recently posed by A. M. Vershik \cite{V}. We prove that for any "generic" metric the number of $(n-m)$-dimensional faces, $0\leq m\leq n$, equals $\binom{n+m}{m,m,n-m}=(n+m)!/m!m!(n-m)!$. This fact is intimately related to regular triangulations of the root polytope (the convex hull of the roots of $A_n$ root system). Also we get two-sided estimates for the logarithm of the number of Vershik classes of metrics: $n^3\log n$ from above and $n^2$ from below.