Resumen de Sharp quantitative isoperimetric inequalities in the $L^1$ Minkowski plane

Benoît Kloeckner

• An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be.

  The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular by Hall and by Fusco, Maggi and Pratelli, but the case has drawn much less attention.

  In this note we prove two quantitative isoperimetric inequalities in the Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains for a quantitative isoperimetric inequality: the only such known result is for the euclidean plane, due to Alvino, Ferone and Nitsch.