**Marton Naszodi **

Eotvos University, Budapest

#### Thursday, April 19 at 2:30 PM

**Abstract**

According to a recent result of Brazitikos, Chasapis and Hioni, if alpha d points are drawn uniformly from a convex body K in 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 based on the combination of two beautiful results, one from combinatorics, the other from geometry.