Let $V\subset\mathbb{R}^n$ be a $k$-dimensional subspace and $c>0$. We denote by $P$ and $Q$, respectively, the orthogonal projection on $V$ and $V^{\perp}$. And we denote by $C(V,c)$ the set $\{ x\in \mathbb{R}^n \mid |Q(x)| \le c |P(x)| \}$. Now we fix a $c>0$. How is the following statement showed?
There exists a finite collection of $k$-dimensional linear subspaces $\{ V_1, \dots, V_N\}$ such that for every $k$-dimensional linear subspace $V$ we have $C(V,c)\subset C(V_m, 2c)$ for some $V_m$.