Lieb-Thirring inequality with semiclassical constant and gradient error term

In 1975, Lieb and Thirring derived a semiclassical lower bound on the kinetic energy for fermions, which agrees with the Thomas-Fermi approximation up to a constant factor. Whenever the optimal constant in their bound coincides with the semiclassical one is a long-standing open question. We prove an improved bound with the semiclassical constant and a gradient error term which is of lower order.