CCDG Seminar

Let $K\subset \mathbb{R}^n$ be a convex body of volume one. Let $X_1,\ldots,X_N$ be independent random vectors distributed uniformly in $K$ and form their convex hull K_N=\mathop{\rm conv}\{ X_1,\ldots, X_N\}. A result of Groemer's states that the expected volume of $K_N$ is smallest when $K$ is the Euclidean ball $B$ of volume one. Similar results hold for other convex sets associated with $K$, e.g., random zonotopes, centroid bodies and their $L_p$-analogs. I will discuss a streamlined approach to inequalities of this type, a key feature being the use of laws of large numbers.