Besicovitch covering theorem

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In mathematical analysis, a Besicovitch cover is an open cover of a subset E of the Euclidean space R^N by balls such that each point of E is the center of some ball in the cover.

The Besicovitch covering theorem asserts that there exists a constant c_N depending only on the dimension N with the following property:

• Given any Besicovitch cover F of a bounded set E, there are c_N subcollections of balls A₁={B_n1}, …, A_cN={B_ncN} contained in F such that each collection A_i consists of disjoint balls, and

[edit] References

• “A general form of the covering principle and relative differentiation of additive functions, I”, Proceedings of the Cambridge Philosophical Society 41: 103-110, 1945 .

• “A general form of the covering principle and relative differentiation of additive functions, II”, Proceedings of the Cambridge Philosophical Society 42: 205-235, 1946 .

• DiBenedetto, E (2002), Real analysis, Birkhäuser, ISBN 0-8176-4231-5 .

• Füredi, Z & Loeb, P.A. (1994), “On the best constant for the Besicovitch covering theorem”, Proceedings of the American Mathematical Society 121 (4): 1063-1073, <http://links.jstor.org/sici?sici=0002-9939(199408)121%3A4%3C1063%3AOTBCFT%3E2.0.CO%3B2-W> .