A dyadic path construction as a geometric realization of the 1-Wasserstein distance on the simplex

The standard n -simplex Δ ⁿ Δ n serves as a natural geometric space for representing discrete probability distributions and compositional data. Quantifying dissimilarity between points P, Q ∈ Δ ⁿ P , Q ∈ Δ n is essential, with metrics respecting ordinal structures being particularly valuable. The 1-Wasserstein distance W₁ W 1 excels in this regard but lacks a direct geometric interpretation as a path length on the simplex. This paper addresses this gap by defining a canonical path from P to Q in an “outside-in” recursive manner using dimension-reducing projections. We prove that the length of this path, under a weighted Euclidean metric with weights a = (a₁, , aₙ) a = ( a 1 , ⋯ , a n ) encoding ordinal spacings, equals the weighted 1-Wasserstein distance W1,a(P, Q) W 1 , a ( P , Q ) . Furthermore, we situate this result within Finsler geometry, demonstrating that W1,a W 1 , a is the geodesic distance induced by a continuous Finsler metric on Δ ⁿ Δ n . The canonical path thus emerges as a non-smooth length-minimizing geodesic.