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Characterization of Cheeger sets for convex subsets of the plane. (English) Zbl 1133.52002

A Cheeger set of a body Ω is a subset that maximizes the ratio of volume to boundary. For dimensional reasons, larger sets have an advantage here: in particular, a Cheeger set must meet the boundary of Ω more than once, as otherwise a larger homothetic subset would exist, yielding a better ratio. On the other hand, it is not hard to see that extending the subset into small corners of Ω must reach a point of diminishing returns, getting less and less volume in return for each increase in boundary. In the plane, it is known that Cheeger sets of convex bodies are unique; in higher dimensions very little is known.

The authors characterize the Cheeger set of a convex planar body as follows. For any Ω, there exists a unique t such that the inner parallel body to Ω at distance t has the same area as the disc of radius t; and the Cheeger set is the Minkowski sum of this inner parallel body and disc. In some cases (known elsewhere in the literature as “calibrable sets”) the Cheeger set is equal to Ω; a simple characterization of such sets (not, as the authors point out, entirely novel) is given.

For convex polygons some additional results are given, including a characterization of polygons whose Cheeger set touches every edge. For such polygons, it is possible to compute the volume-to boundary ratio directly from the area, perimeter, and angles of the polygon. Finally, some rather nice counterexamples are given to show that in the absence of convexity all bets are off.

The reader should note that twice in the first paragraph “|δΩ|” appears in error, where “Ω ¯” was apparently intended.


MSC:
52A40Inequalities and extremum problems (convex geometry)
49Q20Variational problems in a geometric measure-theoretic setting
28A75Length, area, volume, other geometric measure theory