Journal of the European Mathematical Society


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Volume 13, Issue 1, 2011, pp. 185–206
DOI: 10.4171/JEMS/248

A sharp isoperimetric inequality in the plane

Angelo Alvino[1], Vincenzo Ferone[2] and Carlo Nitsch[3]

(1) Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo 4, 80126, NAPOLI, ITALY
(2) Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo 4, 80126, NAPOLI, ITALY
(3) Dipartimento di Matematica e Applicazioni, Università degli Studi di Napoli Federico II, Via Cintia, Monte S. Angelo 4, 80126, NAPOLI, ITALY

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.

Keywords: Isoperimetric inequality, Bonnesen-style inequality, Fraenkel asymmetry, isoperimetric deficit

Alvino Angelo, Ferone Vincenzo, Nitsch Carlo: A sharp isoperimetric inequality in the plane. J. Eur. Math. Soc. 13 (2011), 185-206. doi: 10.4171/JEMS/248