Applied Analysis Seminar The quantitative isoperimetric inequality: A calibration argument

I discuss a novel argument to derive a quantitative form of the classical isoperimetric inequality, which in particular implies all previously known versions. The argument is based solely upon a notion of „quantitative calibrations“ from which one may define a coercive relative energy, penalizing the difference between two interfaces in a, e.g., tilt-excess type way. I give some insights how this allows to prove the following local result: Let F ⊂ ℝ^d be a set of finite perimeter with same barycenter and volume as the unit ball B₁(0), then one can bound from below the energy gap Per(F) − Per(B₁) by the associated relative energy (to be defined in the talk) provided this relative energy is sufficiently small. Based on this local version, we then deduce by a simple compactness argument a corresponding result for arbitrary admissible competitors. This is joint work with Tim Laux.