---
abstract: |
    We show that there exists a sequence $\eps_n \searrow 0$ for which the following holds: Let $K \subset \RR^n$ be a compact, convex set with a non-empty interior. Let $X$ be a random vector that is distributed uniformly in $K$. Then there exist a unit vector $\theta$ in $\RR^n$, $t_0 \in \RR$ and $\sigma > 0$ such that $$\sup_{A \subset \RR} \left| \, Prob \, \{ \, \langle X, \theta
    \rangle \in A \, \} \, - \, \frac{1}{\sqrt{2 \pi \sigma}} \int_A
    e^{-\frac{(t - t_0)^2}{2 \sigma^2}} dt \, \right| \leq \eps_n,$$ where the supremum runs over all measurable sets $A
    \subset \RR$, and where $\langle \cdot, \cdot \rangle$ denotes the usual scalar product in $\RR^n$. Furthermore, under the additional assumptions that the expectation of $X$ is zero and that the covariance matrix of $X$ is the identity matrix, we may assert that most unit vectors $\theta$ satisfy ($\ast$), with $t_0 = 0$ and $\sigma = 1$. Corresponding principles also hold for multi-dimensional marginal distributions of convex sets.
author:
- 'B. Klartag[^1]'
title: A Central Limit Theorem for Convex Sets
---

Introduction {#section1}
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We begin with an example. Let $n \geq 1$ be an integer, and consider 