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2016•ESAIM Control Optimisation and Calculus of VariationsOpen access

On the quantitative isoperimetric inequality in the plane

Chiara Bianchini, Gisella Croce, Antoine Henrot

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Abstract

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set Ω, different from a ball, which minimizes the ratio δ(Ω) /λ2(Ω), where δ is the isoperimetric deficit and λ the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

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What this paper is about

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set Ω, different from a ball, which minimizes the ratio δ(Ω) /λ2(Ω), where δ is the isoperimetric deficit and λ the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

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Available abstract

In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set Ω, different from a ball, which minimizes the ratio δ(Ω) /λ2(Ω), where δ is the isoperimetric deficit and λ the Fraenkel asymmetry, giving a new proof of the quantitative isoperimetric inequality. Some new properties of the optimal set are also shown.

Key concepts: Isoperimetric inequality, Isoperimetric dimension, Inequality, Mathematics, Ball (mathematics), Plane (geometry), Set (abstract data type), Asymmetry

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