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 $p\in [1,+\mathrm{\infty })$ in the case of a general Polish space. In particular it avoids the “glueing of couplings” procedure used in most textbooks on optimal transport.