\langle X, \theta \rangle \, , \,
Z \, \right)
\leq \eps_n,$$ where $Z \sim N(0,1)$ is a standard normal random variable. \[thm\_basic\]

We have the bounds $\eps_n \leq C \left( \frac{\log \log
(n+2)}{\log (n+1)} \right)^{1/2}$ and $\delta_n \leq \exp \left(
-c n^{0.99} \right)$ for $\eps_n$ and $\delta_n$ from Theorem \[thm\_basic\], where $c, C >0$ are universal constants. The quantitative estimate we provide for $\eps_n$ is rather poor. While Theorem \[thm\_basic\] seems to be a reasonable analog of the classical central limit theorem for the category of log-concave densities, we are still lacking the precise Berry-Esseen type bound. A plausible guess might be that the logarithmic dependence should be replaced by a power-type decay, in the bound for $\eps_n$.

Theorem \[thm\_basic\] implies the result stated in the abstract of this paper, which does not require isotropicity; indeed, recall that any log-concave density can be made isotropic by applying an appropriate affine map. Thus, any log-concave density in high dimension has at least one almost-gaussian marginal. When the log-concave density is also isotropic, we can assert that, in fact, the vast majority of its marginals are approximately gaussian. An inherent feature of Theorem \[thm\_basic\] is that it does not provide a 