  Suppose X is a random vector, that is distributed 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. A well-understood
situation, is when X is distributed 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> is distributed
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.
