TITLE: History of the high-dimensional volume paradox
QUESTION [30 upvotes]: 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.

REPLY [44 votes]: 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.