A sharp isoperimetric inequality in the plane involving Hausdorff distance

Abstract

We show that among all the convex bounded domain in ℝ2 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.