A covering property of some classes of sets in $$\mathbb{R}^2$$.(English)Zbl 1016.28003

Summary: We prove that if $${\mathcal B}$$ is a class of open bounded subsets of $$\mathbb{R}^n$$ satisfying a simple geometric condition then the following Besicovitch-type covering property is true. For any $$\varepsilon$$ there exists an $$M$$ such that from any subclass $${\mathcal R}\subset{\mathcal B}$$ one can select $$M$$ subclasses of disjoint sets such that the selected sets cover at least the $$1-\varepsilon$$ part of $$\bigcup{\mathcal R}$$.
Thus we get a sufficient geometric condition for the minimal density property and for the $$\text{CV}_q$$ covering properties introduced in [T. Keleti, Real Anal. Exch. 25, No. 1, 33-34 (2000; Zbl 1014.28507)].
During the proof we also get a reverse isoperimetric inequality for the union of star-shaped sets.

MSC:

 28A05 Classes of sets (Borel fields, $$\sigma$$-rings, etc.), measurable sets, Suslin sets, analytic sets 28A75 Length, area, volume, other geometric measure theory

Zbl 1014.28507
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