Comment by jxjnskkzxxhx

Read my conversation with the other person. If you sample N times you'll be able to put a bound on the ratio, and that is the same regardless of dimension. To your example: if the ratio between sets in D=50 is 1e-50 and you ask how long it takes that your estimate bound doesn't contain zero, that will take a long time. Now if I ask you to estimate the ratio between the 2D circle and the 2D square with 50 decimal place, it will take the same time. Therefore, dimension doesn't enter here. This is a general property of Monte Carlo.