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.

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    $\begingroup$ $(1,0)$ and $(0,1)$ is meaningless in $n$ dimensions. $\endgroup$ – Yves Daoust Feb 14 '20 at 10:48
  • $\begingroup$ @YvesDaoust : I changed it to what I am really trying to ask, thank you. $\endgroup$ – jimjim Feb 14 '20 at 10:53
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    $\begingroup$ Obviously, $\sqrt n$ diverges. But the volume of a unit cube is… unit. $\endgroup$ – Yves Daoust Feb 14 '20 at 10:53

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.


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