7/30/2019 Volume of a Ball in N Dimensions

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Volume of a Ball in N Dimensions

The unit ball in Rn

is defined as the set of points (x1,...,xn) such that

x12 + ... + xn2

7/30/2019 Volume of a Ball in N Dimensions

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Gamma(x+1) by xx

e-x

(2 Pi x)1

for large x, to see why the surprising fact above is true.

Another heuristic is the following probabilistic argument. Pick n points independently and

identically distributed (i.i.d.) from a uniform distribution in [-1,1], and form an n-tuple out ofthese numbers. The resulting vector represents a point picked randomly out of the unit box

B=[-1,1]n, so the probability that such a point is in the unit n-ball is the ratio R(n) of thevolume V(n) to the volume of the unit box, which is 2

n.

Notice that if there are just two coordinates of this point that are greater than 1/Sqrt[2], then

the point cannot be in the unit n-ball. As n grows, we choose more and more coordinates i.i.d.from the uniform distribution, and the smaller the probability is that just zero or one of those

n coordinates are bigger than 1/Sqrt[2]. A little thought reveals that for large n, this

probability decreases by about 1/Sqrt[2] for each new coordinate that is chosen. This shows

that the ratio R(n) tends to 0 as n goes to infinity.

However, we hope to show that V(n)=2nR(n) tends to 0 as n goes to infinity. A refinement of

the above argument will do the trick: if there are just five coordinates of this point that aregreater than 1/Sqrt[5], then the point cannot be in the unit n-ball. For large n, as each new

coordinate chosen, theprobabilitythan less than five coordinates are bigger than 1/Sqrt[5]

drops by about 1/Sqrt[5]. So V(n) changes by about a factor 1/Sqrt[5] as n is incremented, forlarge n. On the other hand, the factor 2n changes by a factor of 2 as n is incremented, for large

n. Hence 2n

changes by a factor of 2/Sqrt[5] for large enough n, so whatever this quantity is, it

eventually gets smaller and smaller.

http://www.math.hmc.edu/cgi-bin/funfacts/main.cgi?Subject=00&Level=0&Keyword=probabilityhttp://www.math.hmc.edu/cgi-bin/funfacts/main.cgi?Subject=00&Level=0&Keyword=probabilityhttp://www.math.hmc.edu/cgi-bin/funfacts/main.cgi?Subject=00&Level=0&Keyword=probabilityhttp://www.math.hmc.edu/cgi-bin/funfacts/main.cgi?Subject=00&Level=0&Keyword=probability