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

Chiara Bianchini (1) , Gisella Croce (2) , Antoine Henrot (3)

1 Dipartimento di Matematica "Ulisse Dini"
2 LMAH - Laboratoire de Mathématiques Appliquées du Havre
3 IECL - Institut Élie Cartan de Lorraine

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.

Mots clés: Isoperimetric inequality; quantitative isoperimetric inequality; isoperimetric deficit; barycentric asymmetry; shape derivative; optimality conditions

Domaines: Equations aux dérivées partielles [math.AP]

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