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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Sharp quantitative isoperimetric inequalities in the $ L^1$ Minkowski plane


Author: Benoît Kloeckner
Journal: Proc. Amer. Math. Soc. 138 (2010), 3671-3678
MSC (2010): Primary 51M16, 51M25; Secondary 49Q20, 52A60
Published electronically: April 26, 2010
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Abstract: An isoperimetric inequality bounds from below the perimeter of a domain in terms of its area. A quantitative isoperimetric inequality is a stability result: it bounds from above the distance to an isoperimetric minimizer in terms of the isoperimetric deficit. In other words, it measures how close to a minimizer an almost optimal set must be.

The euclidean quantitative isoperimetric inequality has been thoroughly studied, in particular by Hall and by Fusco, Maggi and Pratelli, but the $ L^1$ case has drawn much less attention.

In this note we prove two quantitative isoperimetric inequalities in the $ L^1$ Minkowski plane with sharp constants and determine the extremal domains for one of them. It is usually (but not here) difficult to determine the extremal domains for a quantitative isoperimetric inequality: the only such known result is for the euclidean plane, due to Alvino, Ferone and Nitsch.


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Additional Information

Benoît Kloeckner
Affiliation: Institut Fourier, UMR5582, 100 rue des Maths, BP 74, 38402 St. Martin d’Hères, France
Email: bkloeckn@fourier.ujf-grenoble.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10366-9
PII: S 0002-9939(10)10366-9
Received by editor(s): July 28, 2009
Received by editor(s) in revised form: January 5, 2010, and January 6, 2010
Published electronically: April 26, 2010
Communicated by: Nigel J. Kalton
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.