This is the answer to your question: how many sample points do you need to estimate the volume of some region in high dimensions.

Assume that the set K is contained in some enclosing set B whose volume is known to be 1 without loss of generality. Denote by V the unknown volume of K. Let f(x) denote the indicating function of K: f(x) = 1 if x is in K, and f(x) = 0 if x is not in K. (Apologies, no LaTeX on stackoverflow).

Clearly, V is exactly the expectation E[f(x)], which is also the probability P[f(x)=1], where the random variable x is drawn uniformly at random in B. Furthermore, the variance var(f(x)) is precisely V(1-V), which is bounded between 0 and 0.25.

For x 1 ,...,x n drawn uniformly and independently at random in B, consider the sum S n = (f(x 1 ) + ... + f(x n ))/n. The central limit theorem says that S n is asymptotically a normal distribution N(V,var(f(x))/n), so the standard deviation of S n is sqrt(V(1-V)/n). Therefore, if you want a absolute accuracy of ε (i.e. |S n -V|≤ε) with probability p = 0.999999998027 (i.e. 6 standard deviations), you should take n = 36V(1-V)/ε2.

Especially when V is tiny, you are going to want a relative tolerance, i.e. ε = δV. Then, you need n = 36(1-V)/(Vδ2).

The problem that you see here is that it's very tough to get tight accuracies for the case when V is small, but what makes this bad is the curse of dimensionality, which roughly says that V is likely to be O(10-d) in dimensions d.

For example, if K is the sphere of radius 1/2 inscribed in the cube B of side 1, then V is approximately (πe/(2d))d/2/sqrt(dπ), and if you want a relative error less than 100%, then you're going to need just about n = O(dd/2) samples.