to an isotropic position via an affine map, that is, $f \circ T$ is isotropic for some affine map $T:\RR^n \rightarrow \RR^n$ (see, e.g., [@MP]). Suppose that $X$ and $Y$ are two random variables attaining values in some measure space $\Omega$ (here $\Omega$ will always be $\RR$ or $\RR^n$ or a subspace $E \subset
\RR^n$). We define their total-variation distance as $$d_{TV}(X, Y) =
2  \sup_{A \subset \Omega} \left| \, Prob \left \{ X \in A \right
\} \, - \, Prob \left \{ Y \in A \right \} \, \right|,$$ where the supremum runs over all measurable sets $A \subset
\Omega$. Note that $ d_{TV}(X, Y)$ equals the $L^1$-distance between the densities of $X$ and $Y$, when these densities exist. Let $\sigma_{n-1}$ stand for the unique rotationally-invariant probability measure on $S^{n-1}$, also referred to as the uniform probability measure on the sphere $S^{n-1}$.

There exist sequences $\eps_n \searrow 0,
\delta_n \searrow 0$ for which the following holds: Let $n \geq 1$, and let $X$ be a random vector in $\RR^n$ with an isotropic, log-concave density. Then there exists a subset $\Theta \subset S^{n-1}$ with $\sigma_{n-1}(\Theta) \geq 1 - \delta_n$, such that for all $\theta \in \Theta$, $$d_{TV} \left( \, 