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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.

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    $\begingroup$ I suspect I would have been one of the students Gromov would have observed sitting with a clouded expression. This isn't the first place I've encountered the statement that volume is concentrated in the corners, but I've never seen a precise explanation. I can easily see that the volume of a high-dimensional cube is concentrated near its boundary. And since the boundary of a high-dimensional cube is made of high-dimensional cubes, it's concentrated near their boundaries, etc. So it's clear the volume is concentrated near facets of some dimension, but why dimension 0? Or am I over-interpreting? $\endgroup$
    – Will Orrick
    Commented Jan 1, 2021 at 17:04
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    $\begingroup$ Having looked at more literature, it seems that "concentrated in the corners" means at distance of order $\sqrt{n}$ (as opposed to order $1$, as it would be if it were concentrated near the centers of faces). The precise statement for the unit cube is that most of the volume is concentrated in a thin spherical shell of outer radius $\frac{1}{2}\sqrt{\frac{n}{3}}+\epsilon$. Most of the volume is, in fact, concentrated near facets of dimensions close to $n/3$. $\endgroup$
    – Will Orrick
    Commented Jan 4, 2021 at 16:57
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    $\begingroup$ @WillOrrick "concentrated in the corners" is perhaps even more misleading when you realise that the distribution of distances measured from the nearest corner is identical to the distribution of distances from the centre. And a $\frac{1}{2}\sqrt{\frac{n}{3}}\pm\epsilon$ shell could not be arbitrarily thin, as I suspect $\epsilon<0.087$ would not cover half the volume even for large $n$, though $\epsilon=0.5$ might eventually cover over $99.9\%$. $\endgroup$
    – Henry
    Commented Aug 23, 2021 at 23:34
  • $\begingroup$ @Henry Thanks for pointing that out. I'm not sure anymore what I had in mind with the "$+\epsilon$. Relevant posts are here and here. The latter states the more sensible result that the set $\{x\in\mathbf{R}^n\mid (1-\epsilon)\sqrt{\frac{n}{3}}<\lvert x\rvert<(1+\epsilon)\sqrt{\frac{n}{3}}\}$contains most of the cube's volume. $\endgroup$
    – Will Orrick
    Commented Aug 25, 2021 at 16:37
  • $\begingroup$ @WillOrrick One of the interesting things for me is that, for large $n$, the thickness of the shell where the probability is concentrated is close to constant. The distribution of the distance from the centre (or from the nearest vertex) has a variance of $\frac{1}{48}$ when $n=1$ but rapidly falls towards $\frac1{60}$ as $n$ increases. $2$ standard deviations is about $0.26$ $\endgroup$
    – Henry
    Commented Aug 25, 2021 at 17:44

4 Answers 4

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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."

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    $\begingroup$ That is a beautiful article, and it still leaves so many interesting questions open. Thanks! $\endgroup$
    – Rodrigo A. Pérez
    Commented Apr 26, 2013 at 17:26
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    $\begingroup$ Regarding the first reference in that article: I can imagine almost no better name for the author of a lecture entitled An elementary introduction to modern convex geometry than K. Ball. $\endgroup$
    – cardinal
    Commented Apr 27, 2013 at 2:18
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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.

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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.

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http://en.wikipedia.org/wiki/Curse_of_dimensionality mentions a 1957 paper by Richard Bellman.

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    $\begingroup$ Wasn't Bellman refering to what we now call "combinatorial explosion"? $\endgroup$
    – alvarezpaiva
    Commented Apr 27, 2013 at 8:49

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