Volume of Nested Solids

[PatrickPowers]

Start with one solid properly inside another. Their surfaces do not intersect. As the number of dimensions increases, as long as you have reasonable rules as to how the solids grow the volume of the inner solid will go to zero relative to the outer solid. Cause : as the number of dimensions grow the bulk of the volume grow closer to the surface.

This is an extension of the increasingly misnamed Geometric Problem I Need Help With thread.