  The central limit theorem for convex bodies says that with high probability
the marginal of an isotropic log-concave distribution along a random direction
is close to a Gaussian, with the quantitative difference determined
asymptotically by the Cheeger/Poincare/KLS constant. Here we propose a
generalized CLT for marginals along random directions drawn from any isotropic
log-concave distribution; namely, for $x,y$ drawn independently from isotropic
log-concave densities $p,q$, the random variable $\langle x,y\rangle$ is close
to Gaussian. Our main result is that this generalized CLT is quantitatively
equivalent (up to a small factor) to the KLS conjecture. Any polynomial
improvement in the current KLS bound of $n^{1/4}$ in $\mathbb{R}^n$ implies the
generalized CLT, and vice versa. This tight connection suggests that the
generalized CLT might provide insight into basic open questions in asymptotic
convex geometry.
