Isoperimetric stability of boundary barycenters in the plane
Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 67-80.

Consider an open domain D on the plane, whose isoperimetric deficit is smaller than 1. This note shows that the difference between the barycenter of D and the barycenter of its boundary is bounded above by a constant times the isoperimetric deficit to the power 1/4. This power can be improved to 1/2, when D is furthermore assumed to be a convex domain, in any Euclidean space of dimension larger than 2.

Considérons un domaine planaire ouvert D dont le déficit isopérimétrique est plus petit que 1. Cette note montre que la différence entre le barycentre de D et celui de sa frontière est majoré en norme par le déficit isopérimétrique à la puissance 1/4, à une constante multiplicative près. Cette puissance peut être améliorée en 1/2 quand D est de plus supposé être convexe, dans tout espace euclidien de dimension au moins 2.

Published online:
DOI: 10.5802/ambp.383
Classification: 51M04,  51M25,  51M16,  52A20,  52A40,  41A25
Keywords: Isoperimetric inequality on the plane, isoperimetric deficit, boundary barycenter, convex domains, isoperimetric stability
Laurent Miclo 1

1 Institut de Mathématiques de Toulouse, UMR 5219 Toulouse School of Economics, UMR 5314 Université de Toulouse and CNRS 118, route de Narbonne 31062 Toulouse Cedex 9 FRANCE
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Laurent Miclo. Isoperimetric stability of boundary barycenters in the plane. Annales mathématiques Blaise Pascal, Volume 26 (2019) no. 1, pp. 67-80. doi : 10.5802/ambp.383. https://ambp.centre-mersenne.org/articles/10.5802/ambp.383/

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