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The Quantitative Isoperimetric Inequality for Planar Convex Domains

Carlo Nitsch — 2008

Bollettino dell'Unione Matematica Italiana

We prove that among all the convex bounded domains in ℝ 2 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.

A sharp isoperimetric inequality in the plane

Angelo Alvino;  Vincenzo Ferone;  Carlo Nitsch — 2011

Journal of the European Mathematical Society

We show that among all the convex bounded domain in m a t h b b R 2 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.

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