[vlasev]

Here's another look. If you have variables X_1, ..., X_n that are independent and random from normal distributions, if you want someone to be within 1 standard deviation from the mean in EACH dimension, then you are looking at a probability of that happening equal to about 0.68^n, which becomes really small for even a moderate n.

[fats_tromino]

This is the most succinct and clearest explanation of what's going on. I see this discussed a lot when people talk about the curse of dimensionality. Another very simple example is the example of a n-hypercube with edge length 1/2 embedded in the unit n-hypercube. As n increases, the volume of the unit hypercube is constant (1), whereas the volume of the smaller hypercube is decreasing at an exponential rate.