Consider the natural uniform measure (is it called the Haar measure?) on the set of $(n-k)$-dimensional subspaces of $R^n$. We are given a $d$-dimensional affine subspace $U$ (think of $d, k \ll n$; actually, $d=k=O(\sqrt{n})$ will do) of $R^n$ at distance $1$ from the origin. The question is: What is the measure of $(n-k)$-dimensional subspaces $V$ of $R^n$ so that the distance of $P_V U$ from the origin is at most $1/n^c$ (here $P_V U$ is the projection of $U$ on $V$ and $c>0$ is a given constant which can be assumed to be sufficiently large). I don't need a very precise answer; in particular, is the measure upper bounded by $1/n^{c'}$for some other constant $c'$ depending on $c$?
