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Uniqueness of the Cheeger set of a convex body. (English) Zbl 1167.52005
A Cheeger set of a nonempty open bounded subset $$\Omega\subset{\mathbb R}^N$$ is any set $$G\subseteq\Omega$$ minimizing the ratio $$P(F)/|F|$$, $$F\subseteq\Omega$$, where $$|F|$$ denotes the $$N$$-dimensional volume of $$F$$ and $$P(F)$$ the measure of its boundary.
In this paper the authors show that convexity of the set $$\Omega$$ implies the uniqueness of the Cheeger set, which will be, in addition, convex and of class $$C^{1,1}$$. Due to former results the main point here is the uniqueness, which has been known only in the planar case or under additional assumptions. In the planar case, the unique Cheeger set has the nice geometric property to been just the outer parallel body of a certain inner parallel body of $$\Omega$$ [B. Kawohl and T. Lachand-Robert, Pac. J. Math. 225, No. 1, 103–118 (2006; Zbl 1133.52002)].

##### MSC:
 52A20 Convex sets in $$n$$ dimensions (including convex hypersurfaces) 49Q20 Variational problems in a geometric measure-theoretic setting 35J60 Nonlinear elliptic equations 52A38 Length, area, volume and convex sets (aspects of convex geometry)
##### Keywords:
Cheeger set; convex set; set of finite perimeter; mean curvature
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