# On the quantitative isoperimetric inequality in the plane with the barycentric distance

Abstract : In this paper we study the following quantitative isoperimetric inequality in the plane: $\lambda_0^2(\Omega) \leq C \delta(\Omega)$ where $\delta$ is the isoperimetric deficit and $\lambda_0$ is the barycentric asymmetry. Our aim is to generalize some results obtained by B. Fuglede in \cite{Fu93Geometriae}. For that purpose, we consider the shape optimization problem: minimize the ratio $\delta(\Omega)/\lambda_0^2(\Omega)$ in the class of compact connected sets and in the class of convex sets.
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https://hal.archives-ouvertes.fr/hal-02090603
Contributor : Gisella Croce <>
Submitted on : Monday, July 26, 2021 - 2:24:32 PM
Last modification on : Wednesday, July 28, 2021 - 4:04:12 AM

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• HAL Id : hal-02090603, version 2

### Citation

Chiara Bianchini, Gisella Croce, Antoine Henrot. On the quantitative isoperimetric inequality in the plane with the barycentric distance. 2021. ⟨hal-02090603v2⟩

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