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

Benoit Kloeckner (1)

1 IF - Institut Fourier

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.

Mots clés: quantitative isoperimetric inequality; Minkowski plane

Domaines: Analyse fonctionnelle [math.FA]; Géométrie différentielle [math.DG]

Benoit Kloeckner. Sharp quantitative isoperimetric inequalities in the $L^1$ Minkowski plane. Proceedings of the American Mathematical Society, 2010, 138, pp.3671-3678. ⟨10.1090/S0002-9939-10-10366-9⟩. ⟨hal-00408089⟩