[yaakov34]

Actually, you have that exactly right, and it's a very important fact in mathematics and statistics. A unit sphere takes up a smaller and smaller part of a unit cube as the dimension grows (and a unit cone is similar). In other words, a unit circle fills up most of the unit square (~3.14 out of 4), a unit sphere fills a little over half of the unit cube (~4.2 out of 8), and as the dimension grows, the fraction becomes negligible.

Imagine that you have something which depends on many variables (hundreds), and you're trying to predict its behavior based on your previous experience. There is a high chance that the next combination of variable values that you see will be in one of the corners of the many-dimensional cube, because that's where the volume is (the central part of the cube has negligible volume, as we said above). This means that every measurement is in effect an outlier along several dimensions, making predictions very difficult. This is part of the "curse of dimensionality" in statistics. I have seen some people with excellent understanding of mathematics trip themselves up in this area.

[quickthrower2]

Thanks, that is a highly intuitive explanation