I'm interested in the properties of randomly generated convex shapes in $n$-dimensional space.
Suppose I were to generate $v$ uniformly distributed random points on the $n$-ball of radius $R$. What is the probability that their convex hull contains the $n$-ball of radius $1$? There seems to be some knowledge of the probability that their convex hull contains the origin, which provides a handy upper bound, but I'd prefer some nontrivial lower bound. I've googled around for it and haven't found any clues.
Alternatively, given these vertices, is there some reasonably swift test I could perform on them to decide whether or not their hull contains the unit ball? An estimation of this probability from experimental data would be good enough for what I'm doing.