Sharp quantitative isoperimetric inequalities in the 𝐿¹ Minkowski plane

Kloeckner, Benoît

doi:10.1090/S0002-9939-10-10366-9

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

Proc. Amer. Math. Soc. 138 (2010), 3671-3678

DOI: https://doi.org/10.1090/S0002-9939-10-10366-9

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.

References

A. Alvino, V. Ferone, and C. Nitsch. A sharp isoperimetric inequality in the plane, to appear in J. Eur. Math. Soc.
Angelo Alvino, Vincenzo Ferone, and Carlo Nitsch, A sharp isoperimetric inequality in the plane involving Hausdorff distance, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 20 (2009), no. 4, 397–412. MR 2550854, DOI 10.4171/RLM/555
Luigi Ambrosio and Paolo Tilli, Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, vol. 25, Oxford University Press, Oxford, 2004. MR 2039660
Felix Bernstein, Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene, Math. Ann. 60 (1905), no. 1, 117–136 (German). MR 1511289, DOI 10.1007/BF01447496
John E. Brothers and Frank Morgan, The isoperimetric theorem for general integrands, Michigan Math. J. 41 (1994), no. 3, 419–431. MR 1297699, DOI 10.1307/mmj/1029005070
T. Bonnesen. Les problèmes des isopérimètres et des isépiphanes. Gauthier-Villars, Paris, 1929.
N. Fusco, F. Maggi, and A. Pratelli, The sharp quantitative isoperimetric inequality, Ann. of Math. (2) 168 (2008), no. 3, 941–980. MR 2456887, DOI 10.4007/annals.2008.168.941
A. Figalli, F. Maggi, and A. Pratelli. A mass transportation approach to quantitative isoperimetric inequalities. Preprint, 2009.
R. R. Hall, A quantitative isoperimetric inequality in $n$-dimensional space, J. Reine Angew. Math. 428 (1992), 161–176. MR 1166511, DOI 10.1515/crll.1992.428.161
Carlo Nitsch, The quantitative isoperimetric inequality for planar convex domains, Boll. Unione Mat. Ital. (9) 1 (2008), no. 3, 573–589. MR 2455332
Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 500557, DOI 10.1090/S0002-9904-1978-14553-4
Carla Peri, Jörg M. Wills, and Andreana Zucco, On Blaschke’s extension of Bonnesen’s inequality, Geom. Dedicata 48 (1993), no. 3, 349–357. MR 1254104, DOI 10.1007/BF01264078
Jean E. Taylor, Crystalline variational problems, Bull. Amer. Math. Soc. 84 (1978), no. 4, 568–588. MR 493671, DOI 10.1090/S0002-9904-1978-14499-1