We show that among all the convex bounded domain in ℝ2 having an assigned Fraenkel asymmetry index, 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. J. Eur. Math. Soc. 13 (2011), no. 1, pp. 185–206DOI 10.4171/JEMS/248