  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.
