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Register a new userSharp quantitative isoperimetric inequalities in the $L^1$ Minkowski plane
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Authors
Benoit Kloeckner
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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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We prove the sharp quantitative stability for a wide class of weighted isoperimetric inequalities. More precisely, we consider isoperimetric inequalities in convex cones with homogeneous weights. Inspired by the proof of such isoperimetric inequalities through the ABP method, we construct a new convex coupling (i.e., a map that is the gradient of a convex function) between a generic set $E$ and the minimizer of the inequality (as in Gromovs proof of the isoperimetric inequality). Even if this map does not come from optimal transport, and even if there is a weight in the inequality, we adapt the methods of Figalli-Maggi-Pratelli and prove that if $E$ is almost optimal for the inequality then it is quantitatively close to a minimizer up to translations. Then, a delicate analysis is necessary to rule out the possibility of translations. As a step of our proof, we establish a sharp regularity result for restricted convex envelopes of a function that might be of independent interest.
In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $Omega$, different from a ball, which minimizes the ratio $delta(Omega)/lambda^2(Omega)$, where $delta$ is the isoperimetric deficit and $lambda$ the Fraenkel asymmetry, giving a new proof ofthe quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.
We formulate an isoperimetric deformation of curves on the Minkowski plane, which is governed by the defocusing mKdV equation. Two classes of exact solutions to the defocusing mKdV equation are also presented in terms of the $tau$ functions. By using one of these classes, we construct an explicit formula for the corresponding motion of curves on the Minkowski plane even though those solutions have singular points. Another class give regular solutions to the defocusing mKdV equation. Some pictures illustrating typical dynamics of the curves are presented.
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