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The volume of a $d$ dimensional hypersphere of radius $r$ is given by:

$$V(r,d)=\frac{(\pi r^2)^{d/2}}{\Gamma(\frac{d}{2}+1)}$$

What intrigues me about this, is that $V\to 0$ as $d\to\infty$ for any fixed $r$. How can this be? For fixed $r$, I would have thought adding a dimension would make the volume bigger, but apparently it does not. Anyone got a good explanation?

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Think about the square of side length $1/2$. What is its volume in $\mathbb{R}^d$? –  JavaMan Sep 23 '11 at 18:53
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Imagine a square of side length $\frac{1}{2}$: the area is $\frac{1}{4}$. A cube of edge length $\frac{1}{2}$ has volume $\frac{1}{8}$. A hypercube of edge length $\frac{1}{2}$ has hypervolume $\frac{1}{16}$; etc. For a cube, whether going to the "next dimension" increases the total hypervolume or decreases it (as a scalar) depends on whether the edge side was less than $1$ or more than $1$. That suggests to me that "adding a dimension" does not, in and of itself, necessarily mean "making the volume bigger"... –  Arturo Magidin Sep 23 '11 at 18:55
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What about a sphere of radius $10^{10}$? The volume eventually goes to zero :-) –  probabilityislogic Sep 23 '11 at 19:03
    
@probabilityislogic: The point is that simply "adding a dimension makes things bigger" doesn't work: it's not just about the dimension, there are other factors at play. –  Arturo Magidin Sep 23 '11 at 19:26
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Shouldn't the title include the phrases hypershpere and n-->inf? –  ripper234 Oct 4 '11 at 23:19
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6 Answers

up vote 30 down vote accepted

I suppose you could say that adding a dimension "makes the volume bigger" for the hypersphere, but it does so even more for the unit you measure the volume with, namely the unit cube. So the numerical value of the volume does go towards zero.

Really, of course, it is apples to oranges because volumes of different dimensions are not commensurable -- it makes no sense to compare the area of the unit disk with the volume of the unit sphere.

All we can say is that in higher dimensions, a hypersphere is a successively worse approximation to a hypercube (of side length twice the radius). They coincide in dimension one, and it goes downward from there.

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This is a very nice explanation. Thanks :) –  Srivatsan Sep 23 '11 at 21:27
    
This is nice and concise. Here is somebody's blog post saying the same thing with more words. –  yasmar Sep 24 '11 at 10:27
    
You might also want to read this similar post on MSE math.stackexchange.com/questions/15656/… –  Tpofofn Sep 24 '11 at 15:38
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The reason is because the length of the diagonal cube goes to infinity.

The cube in some sense does exactly what we expect. If it's side lengths are $1$, it will have the same volume in any dimension. So lets take a cube centered at the origin with side lengths $r$. Then what is the smallest sphere which contains this cube? It would need to have radius $r\sqrt{d}$, so that radius of the sphere required goes to infinity.

Perhaps this gives some intuition.

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This is the clearest answer for me. –  Sklivvz Sep 24 '11 at 7:56
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+1 Simple, nice, and a clear example. –  Fixed Point Aug 23 '13 at 4:30
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This was thoroughly discussed on MO. I quote from the top answer:

The ultimate reason is, of course, that the typical coordinate of a point in the unit ball is of size $\frac{1}{\sqrt{n}}\ll 1$. This can be turned into a simple geometric argument (as suggested by fedja) using the fact that an $n$-element set has $2^n$ subsets:

At least $n/2$ of the coordinates of a point in the unit ball are at most $\sqrt{\frac{2}{n}}$ in absolute value, and the rest are at most $1$ in absolute value. Thus, the unit ball can be covered by at most $2^n$ bricks (right-angled parallelepipeds) of volume $$\left(2\sqrt{\frac{2}{n}}\right)^{n/2},$$ each corresponding to a subset for the small coordinates. Hence, the volume of the unit ball is at most $$2^n \cdot \left(2\sqrt{\frac{2}{n}}\right)^{n/2} = \left(\frac{128}{n}\right)^{n/4}\rightarrow0.$$ In fact, the argument shows that the volume of the unit ball decreases faster than any exponential, so the volume of the ball of any fixed radius also goes to $0$.

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I'm not seeing why the bricks each have volume at most $\left(2\sqrt{\frac{2}{n}}\right)^{n/2}.$ Shouldn't this be multiplied by a factor of $2^{n/2}$ since the other coordinates have absolute value at most $1$? This doesn't change the limit obviously, but some enlightenment would be nice. –  lyj Feb 14 '13 at 8:30
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Let $X_j$ be a sequence of independent random variables with uniform distribution in the interval $[-r,r]$ (i.e. you are picking an infinite sequence of random numbers from the interval). The probability that $(X_1, \ldots, X_d)$ lies in your hypersphere, i.e. that $R_d = X_1^2 + \ldots + X_d^2 \le r^2$, is $V(r,d)/(2r)^d$ (which by scaling doesn't depend on $r$, so let's call it $P(d)$). Of course $P(d) \to 0$ as $d \to \infty$, but the point is that it goes to 0 faster than an exponential in $d$. Indeed, by the theory of large deviations, for any $t > 0$ we should have $P(R_d \leq t d) \approx e^{-d I(t)}$ as $d \to \infty$ for some function $I(t)$, where $I(t) \to \infty$ as $t \to 0+$.

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Came into this thread to give this proof, but I see someone beat me to it. –  Michael Lugo Sep 23 '11 at 20:44
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Consider the distance from the center of the sphere of radius $R$ in $\mathbb{R}^n$ to the corner of its enclosing cube: $R\sqrt{n}$. This is like covering the surface of the sphere with a layer of tall corners.

For $x=(x_1,x_2,\dots,x_n)\in\mathbb{R}^n$, define $$ Q(x)=x\frac{|x|}{\max_i|x_i|}\tag{1} $$ $Q$ maps a sphere to its enclosing cube. As described above, $\frac{|Q(x)|}{|x|}$ reaches a maximum of $\sqrt{n}$ in the corners. To be exact, $$ \frac{|Q(x)|^2}{|x|^2}=\sum_j\frac{|x_j|^2}{\max_i|x_i|^2}\tag{2} $$ By considering each $x_i$ to be normally distributed and using homogeneity of $(2)$, we get that the mean of $\frac{|Q(x)|^2}{|x|^2}$ over the sphere is $\frac{n+2}{3}$. Thus, the rms average of $\frac{|Q(x)|}{|x|}$ is $\sqrt{\frac{n+2}{3}}$. In this way, the volume of the cube should be approximated by $\sqrt{\frac{n+2}{3}}^{\;n}$ times the volume of the sphere. This grows faster than any $R^n$.

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Let me develop on the idea presented by Christoph Pegel in the comments of this post.

The first thing is that the volume depends on the radius. $n$-ball or cube of radius $r$ has $r^n$ times bigger volume than that of radius $1$. One should expect then, that at least for big radii the volume of the ball will increase (as it is in the case of the unit cube $[-1,1]^n$ of volume $2^n$).

However, the thing is that somebody once decided (could they decide otherwise?) that Fubini theorem would be nice and $n$-dimensional volume of $[0,1]^n$ will be the same as $(n+1)$-dimensional volume $[0,1]^n\times [0,1]$ -- even though of course one is much smaller than the other!

That's the key point - volume is invariant under cartesian product with the unit interval - like it or not.

So, as Christoph Pegel says it is reasonable to compare volume of the ball $B^n$ with volume of the cylinder $B^n\times [-1,1]$. The second is of course $2$ times bigger but it is a matter of radius as already discussed.

Note that if we compare $B^{n+1}$ with $B^n\times [-1,1]$, we notice that only the zero section is the same. At level $t$ the ball is smaller with radius $\sqrt{1-t^2}$. That means that its volume is $(1-t^2)^{n/2}$ times smaller! This function converges to zero with $n$ (and so does its integral) and thus there's no wonder that it kills any geometric growth (where multiplicative factor $>1$ is constant).

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In my opinion the key point is that there are two different measures involved. Namely the Hausdorff measures $\mathcal H^n$ and $\mathcal H^{n+1}$. The $n$-ball has positive $\mathcal H^n$-volume, while its $\mathcal H^{n+1}$-volume vanishes, since there is really no $(n+1)$-dimensional interior in an $n$-ball. To compare the volumes $\mathcal H^{n}(B^n)$ and $\mathcal H^{n+1}(B^{n+1})$ under a common measure, we make use of $\mathcal H^{n}(A)=\mathcal H^{n+1}(A\times[0,1])$. –  Christoph Aug 22 '13 at 11:08
    
Could you explain why are you working with a cylinder of height two instead of height one? Shouldn't we compare $B^{n+1}$ with $B^n \times [\textrm{something with length one}]$? Otherwise we do double the volume. Everything else makes sense to me. I just don't understand "but it is a matter of radius as already discussed". –  Fixed Point Aug 23 '13 at 8:53
    
@FixedPoint Just because "height" (diameter) of the unit-ball is $2$. If our default ball has radius $1/2$, then I would work with some unit interval. The difference between those cases is a factor proportional to the radius and dependence radius ~ volume was discussed in the first paragraph. –  savick01 Aug 23 '13 at 9:36
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