In this MathOverflow post on visualizing high-dimensional spaces, Terry Tao states that "the fact that most of the mass of a unit ball in high dimensions lurks near the boundary of the ball can be interpreted as a manifestation of the law of large numbers, using the interpretation of a high-dimensional vector space as the state space for a large number of trials of a random variable."
What exactly does he mean?
The only interpretation I've come up with is the following:
Observing n trials of a random variable (for instance $X \sim U(0,1)$) is equivalent to picking a random point inside an n-dimensional unit cube. For large n, most of the volume of the n-dimensional unit cube is near the edges. Thus our point is likely to be near an edge of the cube, which means one of our trials has a large value. This shows some law of rare events: "given enough trials, a rare event will occur".
However, this interpretation must use an n-cube rather than an n-sphere because the trials are independent. Additionally, our conclusion is not the law of large numbers, which states that the average of our trials will converge to the expected value for large n.
Can someone provide a correct interpretation of Tao's comment?