Regularity of critical polygons for the weighted Cheeger functional

Let $\mathcal{J}$ be the functional on the class $\mathcal{P}_n$ of simple polygons used in the paper. A polygon is a critical point of $\mathcal{J}$ when the limit in the paper's steepest-descent condition is zero for every sufficiently small perturbation, and for $n\in\{3,4\}$ the critical points in $\mathcal{P}_n$ are the regular $n$-gons inscribed in the circle of radius $R_n^*$ centered at the origin. Critical polygon conjecture. The result stated above holds for all $n\geq 3$. The proposition is established for triangles and quadrilaterals in the radial-weight setting; the conjecture extends the coincidence between critical points and global maximizers to every number of polygon vertices.