[yaakov34]

I explained above what happens when the dimension grows - spheres and cones do indeed take up a smaller and smaller portion of their unit cube, eventually having negligible volume. This is important in the context of high-dimensional statistics and so on.

If you want to actually have infinite-dimensional volumes, you can't just assign finite values to them in a simple way, or you will have contradictions such as a certain volume being completely covered by a union of things which have 0 volume. In infinite dimensions, you instead have various measures like the Gaussian measure. Feynman's path integrals are a kind of way to assign a value - called amplitude - to an infinite-dimensional manifold (a kind of "volume") of paths. But that takes us well to the side of the idea of the ratio between cube and inscribed figure volumes.