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It is showed that on a plane with a radial density the Four Vertex Theorem holds for the class of all simple closed curves if and only if the density is constant. But for the class of simple closed curves that are invariant under a rotation about the origin, the Four Vertex Theorem holds for every radial density.

On the four vertex theorem in planes with radial density eφ(r)

On the four vertex theorem in planes with radial density e^φ(r)