According to a recent manuscript of Brazitikos, Chasapis and Hioni, if $\alpha d$ points are drawn uniformly from a convex body $K$ in ${\mathbb R}^d$, then their convex hull $P$ satisfies $\frac{c}{d} K \subseteq P$ with high probability, where $\alpha,c>0$ are universal constants. In 2000, Giannopoulos and Milman proved corresponding estimates for fine approximation, that is, where the number of vertices is large. We present a common generalization of these results with a very simple proof.
On the other hand, Barvinok proved that one can find roughly $\frac{1}{\varepsilon^{d/2}}$ points in $K$ such that their convex hull $P$ satisfies $(1-\varepsilon) K \subseteq P$. We will show an improvement to this result, achieved using a new method, in a joint work with Dmitry Ryabogin and Fedor Nazarov.
We will try to present as much of the proofs as possible.