History of the high-dimensional volume paradox 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.

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.

 A: Consideration of higher-dimensional spheres at least goes back to the 19th century.
In his paper: "Über verschiedene Theoreme aus der Theorie der Punktmengen in einem $n$-fach ausgedehnten stetigen Raume $G_n$. Zweite Mitteilung." Acta Mathematica 7 (1885) 105-124, Cantor uses "$n$-dimensionale Vollkugeln" ($n$-dimensional solid spheres) frequently. His calculation of the volume has first been mentioned in a letter to Felix Klein. See J. W. Dauben: "Georg Cantor His Mathematics and Philosophy of the Infinite", Princeton University Press (1990) p.326:

In a letter to Felix Klein of June 6, 1882, Cantor explained the details of his more accurate determination of the volume of the unit sphere of dimension $n$ in a space of dimension $n + 1$. It was true that the volume was always less than or equal to $2^n\pi$. But equality was true only for $n$ = 1, $n$ = 2.  

A: A related (and to me, when I first saw it, much more surprising) Fun Fact:  Divide the n-dimensional cube in half in each of $n$ dimensions, to create $2^n$ smaller cubes of edge length 1/2.  Inscribe a ball in each of these subcubes, and then construct the smallest ball tangent to each of those (and centered at the center of the original cube) like so:
      (source)
What happens to the diameter of the central ball as $n$ gets large?
This question received much attention at an algebraic K-theory conference in Boulder in the early 1980s, where each new arrival was presented with a multiple choice problem:  Without stopping to compute, is the limit $-1$, $0$, $1/2$, $1$, $10$ or $\infty$?  You were allowed to choose any three answers out of six, and place a bet on whether the right answer was among them.  I can report that an overwhelming majority of algebraic K-theorists reason thusly: the answer can't be negative and can't be greater than 1 (the ball, after all, is obviously contained inside a box of side 1!); therefore it's safe to bet on the set 
$\lbrace 0,1/2,1 \rbrace $.  Feel free to make money off this.
A: Brian Hayes wrote a column on the volume of the $n$-sphere for American Scientist a couple of years ago, available online here.  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."
A: http://en.wikipedia.org/wiki/Curse_of_dimensionality mentions a 1957 paper by Richard Bellman.
