A version of the central limit theorem holds for convex sets

You need to know: Probability, random vector, uniform distribution, Euclidean space ${\mathbb R}^n$ , inner product $(x,y)=\sum\limits_{i=1}^n x_i y_i$ in ${\mathbb R}^n$ , unit vector in ${\mathbb R}^n$ , integration in ${\mathbb R}^n$ , measurable sets, convex set in ${\mathbb R}^n$ , compact set in ${\mathbb R}^n$ , interior of a set, supremum.

Background: A convex body in ${\mathbb R}^n$ is a compact, convex set $K\subset{\mathbb R}^n$ with a non-empty interior.

The Theorem: On 29th April 2006, Boáz Klartag submitted to arxiv a paper in which he proved the existence of a sequence $\epsilon_n$ converging to $0$ for which the following holds: For any convex body $K\subset{\mathbb R}^n$ , there exist a unit vector $\theta$ in ${\mathbb R}^n$ , $t_0\in {\mathbb R}$ , and $\sigma>0$ such that $\sup\limits_{A\subset {\mathbb R}}\left|\text{Prob}\{(X,\theta)\in A\}-\frac{1}{\sqrt{2\pi\sigma}}\int_A e^{-\frac{(t-t_0)^2}{2\sigma^2}}dt\right|\leq \epsilon_n,$ where $X$ is a random vector uniformly distributed in K, and the supremum runs over all measurable sets $A\subset {\mathbb R}$ .

Short context: Let $X$ be a random vector that is distributed uniformly in a cube in ${\mathbb R}^n$ with centre in the origin. Then components $X_1, X_2, \dots, X_n$ of $X$ are independent and identically distributed, and, by the classical central limit theorem, $\frac{1}{\sqrt{n}}\sum\limits_{i=1}^n X_i$ is close to a normal distribution if n is large. Moreover, the same is true for $(X,\theta)$ , under some mild conditions on unit vector $\theta$ . The Theorem states that a similar result holds if $X$ is uniformly distributed in any convex body in ${\mathbb R}^n$ . This is surprising because in this case the components $X_i$ may be far from being independent.