*Submitted Paper*

**Inserted:** 6 apr 2006

**Year:** 2006

**Abstract:**

We prove that if $C \subset *R*^N$ is of class $C^2$ and uniformly convex, then the Cheeger set of $C$ is unique. The Cheeger set of $C$ is the set which minimizes, inside $C$, the ratio perimeter over volume.

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