A sharp isoperimetric inequality in the plane involving Hausdorff distance (Q1039880)

The authors obtain a new Bonnesen-style isoperimetric inequality for planar convex sets having an assigned asymmetry index related to the Hausdorff distance. This inequality is sharp. The \textit{Hausdorff asymmetry index} \(\delta(E)\) of a planar convex set \(E\) is defined as the translative Hausdorff distance of \(E\) from a disk \(D_R\) having the same measure: \[ \delta(E)=\min_{x\in \mathbb{R}^2}d_H(E,D_R(x)), \] where \(D_R(x)\) is the disk centered at \(x\), such that \(|E|=|D_R(x)|\), \(d_H\) denotes the Hausdorff distance and \(\mid . \mid\) the measure. In particular they obtain the following sharp inequality: \[ P(E)^2-4\pi|E| \geq 16\delta(E)^2 \] where \(P(E)\) denotes the perimeter of the set \(E\). They determine the set for which equality is attained and prove that such a set is unique (up to a similarity). The result is based on a new symmetrization technique closely related to the circular symmetrization. This technique is well suited to the bidimensional framework.