A Cheeger set of a nonempty open bounded subset is any set minimizing the ratio , , where denotes the -dimensional volume of and the measure of its boundary.
In this paper the authors show that convexity of the set implies the uniqueness of the Cheeger set, which will be, in addition, convex and of class . 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 [B. Kawohl and T. Lachand-Robert, Pac. J. Math. 225, No. 1, 103–118 (2006; Zbl 1133.52002)].