in certain aspects of the phenomenon represented by the classical central limit theorem.

A function $f: \RR^n \rightarrow [0, \infty)$ is log-concave if $$f(\lambda x + (1 - \lambda) y) \geq f(x)^{\lambda}
f(y)^{1-\lambda}$$ for all $x, y \in \RR^n$ and $0 < \lambda <
1$. That is, $f$ is log-concave when $\log f$ is concave on the support of $f$. Examples of interest for log-concave functions include characteristic functions of convex sets, the gaussian density, and several densities from statistical mechanics. In this manuscript, we consider random vectors in $\RR^n$ that are distributed according to a log-concave density. Thus, our treatment includes as a special case the uniform distribution on an arbitrary compact, convex set with a non-empty interior.

We say that a function $f: \RR^n \rightarrow [0, \infty)$ is isotropic if it is the density of a random vector with zero mean and identity covariance matrix. That is, $f$ is isotropic when $$\int_{\RR^n} f(x) dx = 1, \ \ \ \int_{\RR^n} x f(x) dx = 0 \ \ \text{and} \ \ \
\int_{\RR^n} \langle x, \theta \rangle^2 f(x) dx = |\theta|^2$$ for all $\theta \in \RR^n$. Any log-concave function with $0 < \int f < \infty$ may be brought 