specific unit vector $\theta \in S^{n-1}$ for which $\langle X, \theta \rangle$ is approximately normal. This is inevitable: We clearly cannot take $\theta = (1,0,...,0)$ in the example of the cube above, and hence there is no fixed unit vector that suits all isotropic, log-concave densities. Nevertheless, under additional symmetry assumptions, we can identify a unit vector that always works.

Borrowing terminology from Banach space theory, we say that a function $f: \RR^n \rightarrow \RR$ is unconditional if $$f(x_1,...,x_n) = f(|x_1|,...,|x_n|) \ \ \ \text{for all} \ \ x =
(x_1,...,x_n) \in \RR^n.$$ That is, $f$ is unconditional when it is invariant under coordinate reflections.

There exists a sequence $\eps_n \searrow 0$ for which the following holds: Let $n \geq 1$, and let $f:\RR^n
\rightarrow [0, \infty)$ be an unconditional, isotropic, log-concave function. Let $X = (X_1,...,X_n)$ be a random vector in $\RR^n$ that is distributed according to the density $f$. Then, $$d_{TV} \left( \,
\frac{X_1 + ... + X_n}{\sqrt{n}} \, , \, Z \, \right) \leq \eps_n$$ where $Z \sim N(0,1)$ is a standard normal random variable. \[thm\_uncond\]

We provide the estimate $\eps_n \leq \frac{C}{(\log (n+1))^{1/5}}$ for $\eps_n$ from Theorem \[thm\_uncond\]. Multi-dimensional versions of Theorem \[thm\_basic\] are our next 