[kazinator]

But the former formula already tells you that most of the volume is near the high range of r: that which was to be shown.

The surface area to volume concept adds nothing.

Because the volume of a sphere is proportional to r cubed, you know there is much more volume between r in [0.9, 1.0] than in the same sized interval of r [0.0, 0.1].

You can find the break-even point almost in your head. At what r value is half the volume of a R = 1.0 sphere below that value? Why, that's just the cube root of 1/2 ~= 0.794. So almost half the volume is within 20% of the radius from the surface.

That's far from the claim that almost all the volume is near the surface: half is not almost all, and 20% isn't all that near. However, you can see how it gets nearer and nearer for higher dimensions.

For a ten dimensional sphere, the tenth root of 1/2 is ~ 0.933. So over half the volume of a ten dimensional sphere is within the 7% depth.

[10000truths]

The surface area to volume ratio is just a limit of the shell volume to total volume ratio as the shell thickness goes to zero. So both should asymptotically scale with higher dimensions in the same way.