The following easily stated problem has arisen in my research. However, it's outside my field, and I'm unfamiliar with the literature. I would greatly appreciate any references.

Let $K\subset \mathbb{R}^d$ be a convex polytope, and let $f=f(v)$ be a probability mass function on $K$'s vertices. Assume I draw $n$ vertices $v_1,\dots,v_n$ randomly according to $f$, and let $K_n$ be the convex hull of $\{v_1,\dots,v_n\}$. Note $K_n$ is a random convex polytope that is a subset of $K$.

I would like to estimate or bound the expectation of the volume difference $$ \mathbb{E}\left[\frac{\text{vol}(K \backslash K_n)}{\text{vol}(K)}\right], $$ or estimate the probability $$ \text{Prob}\left[\,\frac{\text{vol}(K \backslash K_n)}{\text{vol}(K)}\;\geq\; t\,\right]. $$ Assume, of course, that $f(v)>0$ for all vertices.

I found a survey, Random Polytopes, Convex Bodies, and Approximation, by Bárány that seems closely related, but the problem set up is slightly different. I suspect this problem has been studied before.