x \in \RR^n ; |x| = 1 \}$ for the unit sphere in $\RR^n$. Suppose that $Y = (Y_1,...,Y_n)$ is a random vector that is distributed uniformly in the ball $\sqrt{n + 2} \, D^{n}$. Then $Y_1,...,Y_n$ are identically-distributed random variables of mean zero and variance one, yet they are not independent. Nevertheless, it was already observed by Maxwell that for any $\theta =
(\theta_1,...,\theta_n) \in
S^{n-1}$, the distribution of the random variable $$\langle \theta, Y \rangle = \sum_{i=1}^n \theta_i Y_i$$ is close to the standard normal distribution, when $n$ is large. See, e.g., [@diaconis] for the history of the latter fact and for more information.

There is a wealth of central limit theorems in probability theory that ensure normal approximation for a sum of many independent, or weakly dependent, random variables. Our first example, that of the cube, fits perfectly into this framework. The approach we follow in this paper relates more to the second example, that of the Euclidean ball, where the “true source” of the gaussian approximation may be attributed to geometry. The geometric condition we impose on the distribution of the random variables is that of convexity. We shall see that convexity may substitute for independence 