Combinatorics of the Lipschitz Polytope

Let ρ be a metric on the set X= { 1 , 2 , ⋯ , n+ 1 }. Consider the n-dimensional polytope of functions f: X→ R, which satisfy the conditions f(n+ 1) = 0 , | f(x) - f(y) | ⩽ ρ(x, y). The question on classifying metrics depending on the combinatorics of this polytope have been recently posed by (Vershik, Arnold Math J 1(1):75–81, 2015). We prove that for any “generic” metric the number of (n- m) -dimensional faces, 0 ⩽ m⩽ n, equals (n+mm,m,n-m)=(n+m)!/m!m!(n-m)!. This fact is intimately related to regular triangulations of the root polytope (convex hull of the roots of An root system). Also we get two-sided estimates for the logarithm of the number of Vershik classes of metrics: n3log n from above and n2 from below.