> in high dimensions, the unit cube is all corners and no interior.
Wouldn't it be rather "all faces" ?
As you say, it is useful to think about these volumes in terms of probabilities. You can pick a random point in a one-million dimensional cube by throwing one million random numbers uniformly between -1 and 1. Being near a corner means that each one of these coordinates is very close to either 1 or -1, which is extremely unlikely. Being near a face means that just one of the coordinates is near 1 or -1, which is almost sure to happen! Thus, essentially the whole volume of the cube is near its boundary.
By a similar reasoning, you can see that the volume of the sphere is negligible with respect to that of the cube. Consider your random point inside the cube, it will fall inside the sphere if x_1^2 + ... x_n^2 < 1, which is extremely unlikely if n is large: you are summing a million numbers between 0 and 1, how likely is it that the sum is smaller than 1? Very unlikely: if just one of these numbers was too close to 1, all the others must be nearly zero.
Wouldn't it be rather "all faces" ?
As you say, it is useful to think about these volumes in terms of probabilities. You can pick a random point in a one-million dimensional cube by throwing one million random numbers uniformly between -1 and 1. Being near a corner means that each one of these coordinates is very close to either 1 or -1, which is extremely unlikely. Being near a face means that just one of the coordinates is near 1 or -1, which is almost sure to happen! Thus, essentially the whole volume of the cube is near its boundary.
By a similar reasoning, you can see that the volume of the sphere is negligible with respect to that of the cube. Consider your random point inside the cube, it will fall inside the sphere if x_1^2 + ... x_n^2 < 1, which is extremely unlikely if n is large: you are summing a million numbers between 0 and 1, how likely is it that the sum is smaller than 1? Very unlikely: if just one of these numbers was too close to 1, all the others must be nearly zero.