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.
Cite this article
Carlo Nitsch, Angelo Alvino, Vincenzo Ferone, A sharp isoperimetric inequality in the plane involving Hausdorff distance. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 20 (2009), no. 4, pp. 397–412