Consider an isotropic measure \mu on R^d (i.e., centred and whose covariance is the identity) and let X_1,...,X_m be independent, selected according to \mu. If \Gamma=m^{-1/2} \sum_{i=1}^m <X_i,->e_i is the random operator whose rows are X_i/\sqrt{m}, how does the random set \Gamma S^{d-1} typically look like? For example, if the extremal singular values of \Gamma are close to 1, then \Gamma S^{d-1} is "well approximated" by a d-dimensional section of S^{m-1} and vice-versa. But is it possible to give a more accurate description of that set?

I will show that under minimal assumptions on \mu, with high probability and uniformly in t \in S^{d-1}, each vector \Gamma t  inherits the structure of the one-dimensional marginal <X,t> in a strong sense. If time permits I will also present a few applications of this fact. 

Joint work with D. Bartl