A sharp isoperimetric inequality in the plane involving Hausdorff distance

Abstract

We show that among all the convex bounded domain in $R^2$ having an assigned asymmetry index related to Hausdorff distance, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We also show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domain.