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

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.

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