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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

Citations:

Zbl 1014.28507
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