- #36
Nono713
Gold Member
MHB
- 618
- 4
This seems to describe the average distance of a randomly placed $n$-dimensional point in the unit hypercube, from the origin. For what it's worth I had Mathematica churn out the definite integrals for the first few $n$. I was able to get analytical solutions for up to $n = 3$, but the closed-form expression for $n = 3$ is too unwieldy to post here. For $n > 11$, it takes a long time to produce a result:chisigma said:
$\begin{array}{|l|r|}
n &\mathrm{Integral} \\
~ &~ \\
1 & \frac{1}{2} \\
2 &\frac{\sqrt{2} + \sinh^{-1}{1}}{3} = 0.765 \\
3 &0.961 \\
4 &1.12 \\
5 &1.26 \\
6 &1.39 \\
7 &1.50 \\
8 &1.61 \\
9 &1.71 \\
10 &1.81\\
11 &1.90\\
\end{array}$
It seems to grow at a decreasing rate. The integral is relatively well approximated by $\sqrt{\frac{n}{3}}$, and the approximation gets increasingly better as $n$ increases. This result is trivially obtained by taking the square root outside the integral - this is asymptotically valid as $n$ grows large - the Euclidian metric is a poor choice of distance function in high-dimensional space (curse of dimensionality).
I don't see a way to nicely calculate a closed-form expression of the integral with respect to $n$, but I suspect it is possible. That's all I can say though (Sadface)
