The quantitative isoperimetric inequality: A calibration argument

Sebastian Hensel (Uni Leipzig) #

Please note that the seminar will take place on Tuesday, January 27, 14:00-15:00, in Seminar Room 10 (5th floor)!!! #

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 \(F \subset \mathbb{R}^d\) be 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. 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.