The quantitative isoperimetric inequality: A calibration argument

I discuss a novel argument to derive a quantitative form of the isoperimetric inequality. 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 $F \subset \mathbb{R}^d$ a set of finite perimeter with same barycenter and volume as the unit ball $B_1(0)$, then one can bound from below the energy gap $Per(F) - Per(B_1)$ by the associated relative energy (to be defined in the talk) provided this relative energy is sufficiently small. This is joint work with Tim Laux (U Heidelberg). The corresponding "global" result is part of ongoing research.