Isoperimetric stability of boundary barycenters in the plane

Abstract

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. Keywords: Isoperimetric inequality on the plane, isoperimetric deficit, boundary barycenter, convex domains, isoperimetric stability.