Suppose X is a random vector, that distributes uniformly in some n-dimensional convex set. It was conjectured that when the dimension n is very large, there exists a non-zero vector u, such that the distribution of the real random variable <X,u> is close to the gaussian distribution. This conjecture appears explicitly in the works of Anttila, Ball and Perissinaki and of Brehm and Voigt, and may be essentially traced back to Gromov and others. A well-understood situation, is when X distributes uniformly over the n-dimensional cube. In this case, <X,u> is approximately gaussian for, say, the vector u = (1,...,1) / sqrt(n), as follows from the classical central limit theorem.

We prove the conjecture for a general convex set. Moreover, when the expectation of X is zero, and the covariance of X is the identity matrix, we show that for "most" unit-vectors u, the random variable <X,u> distributes approximately according to the gaussian law. We argue that convexity - and perhaps geometry in general - may replace the role of independence in certain aspects of the phenomenon represented by the central limit theorem. See also arXiv:math.MG/0605014.