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: ${\lambda }_0^2(\Omega )\le C\delta (\Omega )$ 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.

References in corpus (2)

• Regularity and singularities of Optimal Convex shapes in the plane
• On the maximization of a class of functionals on convex regions, and the characterization of the farthest convex set