  We discuss the Lieb-Thirring inequality for periodic systems, which has the
same optimal constant as the original inequality for finite systems. This
allows us to formulate a new conjecture about the value of its best constant.
To demonstrate the importance of periodic states, we prove that the 1D
Lieb-Thirring inequality at the special exponent $\gamma=3/2$ admits a
one-parameter family of periodic optimizers, interpolating between the
one-bound state and the uniform potential. Finally, we provide numerical
simulations in 2D which support our conjecture that optimizers could be
periodic.
