2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/160002We provide a sharp rate of convergence in the central limit theorem for random vectors with an unconditional, log-concave density. The argument relies on analysis of the Neumann laplacian on convex domains and on the theory of optimal transportation of measures.34 pages, minor revision: Some remarks and explanations were added. To appear in Probab. Theory Related FieldsProbabilityMetric Geometrytext