Besicovitch covering theorem

In mathematical analysis, a Besicovitch cover, named after Abram Samoilovitch Besicovitch, 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

${\displaystyle E\subseteq \bigcup _{i=1}^{c_{N}}\bigcup _{B\in A_{i}}B.}$

Let G denote the subcollection of F consisting of all balls from the c_N disjoint families A₁,...,A_cN. The less precise following statement is clearly true: every point x ∈ R^N belongs to at most c_N different balls from the subcollection G, and G remains a cover for E (every point y ∈ E belongs to at least one ball from the subcollection G). This property gives actually an equivalent form for the theorem (except for the value of the constant).

• There exists a constant b_N depending only on the dimension N with the following property: Given any Besicovitch cover F of a bounded set E, there is a subcollection G of F such that G is a cover of the set E and every point x ∈ E belongs to at most b_N different balls from the subcover G.

In other words, the function S_G equal to the sum of the indicator functions of the balls in G is larger than 1_E and bounded on R^N by the constant b_N,

${\displaystyle \mathbf {1} _{E}\leq S_{\mathbf {G} }:=\sum _{B\in \mathbf {G} }\mathbf {1} _{B}\leq b_{N}.}$