In the case where $E\subset\mathbb{R}^1$, a Besicovitch cover of $E$ is a cover by open intervals such that each point of $E$ is the center of some interval in the cover.

The Besicovitch Covering Theorem says that there exists some $n$ such that for any bounded set $E\subset\mathbb{R}$ with Besicovitch cover $F$, there exist $A_1,\ldots,A_n$ subsets of $F$ such that each is a disjoint collection of intervals and $\bigcup_{i=1}^nA_i$ covers $E$.

I have sketched out a proof that $n=2$ following the remark on page 10 of http://torus.math.uiuc.edu/jms/Papers/thesis/besic-jga.pdf that this is trivial, and would like to use this fact in a paper of mine without pretending that the best constant is unknown, though including the proof would be irrelevant to the rest of the paper.

Does anyone know an appropriate citation or is it so universally accepted that nobody has bothered to publish it?


The statement of the Besicovitch Covering Theorem I've learned is a bit different, see Morgan "Geometric Measure Theory; a beginner's guide". The result you state is stated as a lemma there. As an example, the bound for $\mathbb{R}^2$ (18) is given, referring to Reifenberg, E.R. "A problem on circles" published in Math. Gazette 32 (1948), 290-292

The bound for $\mathbb{R}$ seems obvious to me. Probably this is easy to show by hand. I doubt if anyone will mind if you state this as a remark.


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