MathDB

Let

\mathcal B

be a family of open balls in

\mathbb R^n

and

c<\lambda\left(\bigcup\mathcal B\right)

where

\lambda

is the

n

-dimensional Lebesgue measure. Show that there exists a finite family of pairwise disjoint balls

\{U_i\}^k_{i=1}\subseteq\mathcal B

such that

\sum_{j=1}^k\lambda(U_j)>\frac c{3^n}.

Problem Statement