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

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