the cube $Q = [ -\sqrt{3}, \sqrt{3}]^n
\subset \RR^n$. Suppose that $X = (X_1,...,X_n)$ is a random vector that is distributed uniformly in the cube $Q$. Then $X_1,...,X_n$ are independent, identically-distributed random variables of mean zero and variance one. Consequently, the classical central limit theorem states that the distribution of the random variable $$\frac{X_1 + ... + X_n}{\sqrt{n}}$$ is close to the standard normal distribution, when $n$ is large. Moreover, suppose we are given $\theta_1,...,\theta_n \in \RR$ with $\sum_{i=1}^n \theta_i^2 = 1$. Then under mild conditions on the $\theta_i$’s (such as Lindeberg’s condition, see, e.g., [@feller Section VIII.4]), the distribution of the random variable $$\langle \theta, X \rangle = \sum_{i=1}^n \theta_i X_i$$ is approximately gaussian, provided that the dimension $n$ is large. For background on the classical central limit theorem we refer the reader to, e.g., [@feller] and [@stroock].

Let us consider a second example, no less fundamental than the first. We denote by $| \cdot |$ the standard Euclidean norm in $\RR^n$, and let $\sqrt{n + 2} \, D^{n} = \{ x \in \RR^n ;
|x| \leq \sqrt{n+2} \}$ be the Euclidean ball of radius $\sqrt{n + 2}$ around the origin in $\RR^n$. We also write $S^{n-1} = \{ 