Consider an isotropic measure $\mu$ on $\mathbb R^d$ (i.e., centered and whose covariance is the identity) and let $X_1,...,X_m$ be independent, selected according to $\mu$. If $\Gamma$ is the random operator whose rows are $X_i/\sqrt{m}$, how does the image of the unit sphere under $\Gamma$ typically look like? For example, if the extremal singular values of $\Gamma$ are close to 1, then this random set is "well approximated" by a d-dimensional sphere, 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 a unit vector $t$, each vector $\Gamma t$ inherits the structure of the one-dimensional marginal $\langle X,t\rangle$ in a strong sense. If time permits I will also present a few generalisations of this fact - for an arbitrary indexing set. (A joint work with D. Bartl.)