On the Four Vertex Theorem on planes with radial density $e^{\phi(r)}$

TL;DR

This paper investigates the validity of the Four Vertex Theorem on planes with radial density, showing it holds universally only for constant densities but always for rotationally invariant curves regardless of density.

Contribution

It establishes the precise conditions under which the Four Vertex Theorem applies in planes with radial density, distinguishing between general and rotationally invariant curves.

Findings

Four Vertex Theorem holds for all simple closed curves only if density is constant.

For rotationally invariant curves, the theorem holds for any radial density.

The theorem's applicability depends on the symmetry and density conditions.

Abstract

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.