Looked at https://en.wikipedia.org/wiki/Volume_of_an_n-ball, but can't see how to calculate the volume and surface when $\lim n \to \infty $ , trying to find out if volume and surface converge for infinite dimensional unit sphere and cube.
if there are other known cases for infinite dimensions that converge e.g. for distance function between two point $(0,\dots,0)$ and $(1,\dots,1)$ seem to diverge in infinite dimensions.
Best Answer
From the Wikipedia formulas https://en.wikipedia.org/wiki/Volume_of_an_n-ball, both volume and areas are the ratio of a power $n/2$ of $\pi$ and Gamma of $n/2$. The latter grows much faster, as the factors in a factorial grow linearly.