Comment by jdhwosnhw
5 days ago
Do you know how to program? It’s super easy to write a very simple rejection-based program to estimate the volume of a hypersphere (you can do it in <10 lines with numpy). Try it yourself for dimensionality 50 and see how long it takes before the estimate rises above precisely 0
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.