Face Numbers of Random Polytopes and Angles of Random Simplices

Let $X_1,X_2,\dots ,X_n$ be $n$ random points drawn uniformly at random from the $d$-dimensional unit ball. What is the expected number of $k$-dimensional faces of their convex hull $[X_1,\dots ,X_n]$? Let $Y_1,\dots ,Y_{d+1}$ be $d+1$ random points sampled uniformly at random on the unit sphere in ${\mathbb{R}}^d$. What is the expected sum of solid angles of the simplex with vertices at $Y_1,{\text{ldotyY}}_{d+1}$? We shall review recent results on these and some other questions of stochastic geometry.