2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/74314New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applications to the behavior of Weil-Petersson geodesics are discussed.14 pagesDifferential Geometry32G15, 30F60, 53C22text