On the stability of the polygonal isoperimetric inequality
  We obtain a sharp lower bound on the isoperimetric deficit of a general polygon in terms of the variance of its side lengths, the variance of its radii, and its deviation from being convex. Our technique involves a functional minimization problem on a suitably constructed compact manifold and is based on the spectral theory for circulant matrices. 
Emanuel Indrei and Levon Nurbekyan
math.CA math.DG