Green–Wasserstein inequality on compact surfaces

Abstract Let (M, g) (M, g) be a compact connected two-dimensional Riemannian manifold without boundary. In this note, we answer a question posed by Steinerberger: can one remove the n log n factor in the two-dimensional Green–Wasserstein inequality while keeping the unrenormalized off-diagonal Green term? We show that this is impossible on any compact connected surface: there is no inequality of the same form that holds uniformly over point sets with an O (n^-1/2) O (n − 1 / 2) remainder for all n. We argue by contradiction and combine a second-moment estimate for the random Green energy of i. i. d. samples with the semi-discrete random matching asymptotics of Ambrosio–Glaudo.