topic. For integers $k,n$ with $1 \leq k \leq n$, let $G_{n,k}$ stand for the grassmannian of all $k$-dimensional subspaces in $\RR^n$. Let $\sigma_{n,k}$ be the unique rotationally-invariant probability measure on $G_{n,k}$. Whenever we refer to the uniform measure on $G_{n,k}$, and whenever we select a random $k$-dimensional subspace in $\RR^n$, we always relate to the probability measure $\sigma_{n,k}$ defined above. For a subspace $E \subset \RR^n$ and a point $x \in \RR^n$, let $Proj_E(x)$ stand for the orthogonal projection of $x$ onto $E$. A standard gaussian random vector in a $k$-dimensional subspace $E
\subset \RR^n$ is a random vector $X$ that satisfies $Prob \{ X
\in A \} = (2 \pi)^{-k/2} \int_A \exp(-|x|^2/2) dx$ for any measurable set $A \subset E$.

There exists a universal constant $c > 0$ for which the following holds: Let $n \geq 3$ be an integer, and let $X$ be a random vector in $\RR^n$ with an isotropic, log-concave density. Let $\eps > 0$ and suppose that $1 \leq k
\leq c \eps^2 \frac{\log n}{\log \log n}$ is an integer. Then there exists a subset $\E \subset G_{n,k}$ with $\sigma_{n,k}(\E)
\geq 1 - e^{-c n^{0.99}}$ such that for any $E \in \E$, $$d_{TV} \left( 