A Duality-Based Proof of the Triangle Inequality for the Wasserstein Distances

TL;DR

This paper presents a new proof of the triangle inequality for Wasserstein distances using Kantorovich duality, simplifying the traditional approach by avoiding the glueing of couplings in general Polish spaces.

Contribution

It provides a duality-based proof of the triangle inequality for Wasserstein distances that bypasses the common coupling glueing technique.

Findings

Simplifies the proof of the triangle inequality for Wasserstein distances.

Applies to general Polish spaces and all p ≥ 1.

Avoids the glueing of couplings method used in standard textbooks.

Abstract

This short note gives a proof of the triangle inequality based on the Kantorovich duality formula for the Wasserstein distances of exponent $p\in[1,+\infty)$ in the case of a general Polish space. In particular it avoids the "glueing of couplings" procedure used in most textbooks on optimal transport.