The Quantitative Isoperimetric Inequality for Planar Convex Domains
Carlo Nitsch (2008)
Bollettino dell'Unione Matematica Italiana
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We prove that among all the convex bounded domains in having an assigned Fraenkel asymmetry index, there exists only one convex set (up to a similarity) which minimizes the isoperimetric deficit. We show how to construct this set. The result can be read as a sharp improvement of the isoperimetric inequality for convex planar domains.