Abstract The standard n -simplex ⁿ Δ n serves as a natural geometric space for representing discrete probability distributions and compositional data. Quantifying dissimilarity between points P, Q 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, ₀ (P, Q) W 1, a (P, Q). Furthermore, we situate this result within Finsler geometry, demonstrating that W₁, ₀ W 1, a is the geodesic distance induced by a continuous Finsler metric on ⁿ Δ n. The canonical path thus emerges as a non-smooth length-minimizing geodesic.
Cappelletti-Montano et al. (Wed,) studied this question.