Let $K$ be a convex, full-dimensional, bounded region of $\mathbb{R}^n$. More precisely, there exist two balls of radiuses $0<r<R$ such that the ball of radius $r$ is fully contained inside $K$ and the ball of radius $R$ fully contains $K$.
Now suppose that for any positive integer $N$, I have a black-box that generates $N$ independent random points which are uniformly distributed inside $K$. Given this black-box, what's an efficient algorithm to estimate the volume of $K$?
By efficient algorithm, I mean minimize $N$, the size of the sample of points, to obtain a desired standard error in the volume estimate.
Note: You can assume that $r,R$ and the centers of the corresponding balls are known.