Central limit theorems for Gaussian polytopes
  Choose $n$ random, independent points in $\R^d$ according to the standard normal distribution. Their convex hull $K_n$ is the {\sl Gaussian random polytope}. We prove that the volume and the number of faces of $K_n$ satisfy the central limit theorem, settling a well known conjecture in the field. 
I. Barany and V. H. Vu