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We prove that the cd-index of a convex polytope satisfies a strong monotonicity property with respect to the cd-indices of any face and its link. As a consequence, we prove for d-dimensional polytopes a conjecture of Stanley that the cd-index is minimized on the d-dimensional simplex. Moreover, we prove the upper bound theorem for the cd-index, namely that the cd-index of any d-dimensional polytope with n vertices is at most that of C(n, d), the d-dimensional cyclic polytope with n vertices.doi:10.1007/s002090050480 fatcat:7cwpw3mk7vdgzdu2ohwbye77ba