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

TL;DR

This paper establishes quantitative isoperimetric inequalities in the $L^1$ Minkowski plane, showing that nearly optimal domains are close to squares, with precise measurements of closeness and extremal shapes identified.

Contribution

It provides the first sharp quantitative isoperimetric inequalities in the $L^1$ Minkowski plane, characterizing near-extremal domains as close to squares with explicit extremal cases.

Findings

Domains close to isoperimetric minimizers are near squares in $L^ ext{infty}$ Hausdorff distance.

Closeness is also measured via Fraenkel asymmetry, confirming the stability of the square shape.

Extremal domains are explicitly determined in the $L^ ext{infty}$ Hausdorff distance setting.

Abstract

We prove that a plane domain which is almost isoperimetric (with respect to the $L^1$ metric) is close to a square whose sides are parallel to the coordinates axis. Closeness is measured either by $L^\infty$ Haussdorf distance or Fraenkel asymmetry. In the first case, we determine the extremal domains.