Inscribe an $n$-ball in an $n$-dimensional hypercube of side equal to 1, and let $n \rightarrow \infty$. The hypercube will always have volume 1, while it is a fun folk fact (FFF) that the volume of the ball goes to 0.


I first learnt of this in relation to Gromov. In the story I heard, he used to ask incoming students to compute the distance $(\sqrt{n}-1)/2$ from a hypercube corner to the ball, and observe them to see if they realized that the volume of the hypercube is concentrated in its corners.



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> Is this story correct? And is this the origin of this FFF? I could imagine a situation where several people noticed this at different times, but where the fact did not become "viral" until much more recenttly.
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History of the high-dimensional volume paradox
Brian Hayes wrote a column on the volume of the $n$-sphere for *American Scientist* a couple of years ago, available [online here.](https://www.americanscientist.org/article/an-adventure-in-the-nth-dimension) It includes a bit of history, with bibliography, toward the end, which might be of help here.


**Added 4/26/13**: Here are a couple of pertinent passages from Brian's article:


"... Sommerville mentions the
Swiss mathematician Ludwig Schläfli
as a pioneer of n-dimensional geometry.
Schläfli’s treatise on the subject, written
in the early 1850s, was not published
in full until 1901, but an excerpt translated into English by Arthur Cayley appeared in 1858. The first paragraph of
that excerpt gives the volume formula
for an n-ball, commenting that it was
determined “long ago.” An asterisk
leads to a footnote citing papers published in 1839 and 1841 by the Belgian
mathematician Eugène Catalan."


and


"Not one of these early works pauses to comment on the implications of
the formula—the peak at n=5 or the
trend toward zero volume in high dimensions. Of the works mentioned by
Sommerville, the only one to make these
connections is a thesis by Paul Renno
Heyl, published by the University of
Pennsylvania in 1897."


