We consider uniform convex planar regions and lines through their center of mass and lying in the same plane as the region; each line is parametrized by an angle $\alpha$ it makes with some reference direction in that same plane.

Are there uniformly dense convex planar regions other than the circular disk with the property: the region has equal moment of inertia about every line through the center of mass and lying within a finite range of values of $\alpha$? For the circular disk, this range is obviously, the full [0,2$\pi$].

And what could be said about other moments?