A flow approach to the L−2 Minkowski problem

We prove that the set of smooth, π-periodic, positive functions on the unit sphere for which the planar L−2 Minkowski problem is solvable is dense in the set of all smooth, π-periodic, positive functions on the unit sphere with respect to the L∞ 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 special linear transformations.