  We prove that the set of smooth, $\pi$-periodic, positive functions on the
unit circle for which the $L_{-2}$ Minkowski problem is solvable is dense in
the set of all smooth, $\pi$-periodic, positive functions on the unit circle
with respect to the $L^{\infty}$ norm. Furthermore, we obtain a necessary
condition on the solvability of the even $ L_{-2}$ Minkowski problem. At the
end, we prove uniqueness of the solutions up to an affine linear
transformation.
