One way to phrase the
"[concentration-of-measure](https://en.wikipedia.org/wiki/Concentration_of_measure)"
phenomenon is that, 
for a Euclidean sphere $S^d$ in $d$ dimensions, for large $d$,
"most of the mass is close to the equator, for any equator."1



> 
> ***Q***. How could one explain/justify this intuitively—perhaps just verbally—to a mathematically literate but naive audience
>  (say, advanced undergraduate math majors)?
> 
> 
> 


That "most of the mass is close to the equator, for any equator"
seems almost contradictory (imagining orthogonal equatorial hyperplanes), or at the
least, superficially quite puzzling.
Can one only gain intuition via working
through details of the [Brunn–Minkowski theorem](https://en.wikipedia.org/wiki/Brunn%E2%80%93Minkowski_theorem)
or the [isoperimetric inequality](https://en.wikipedia.org/wiki/Isoperimetric_inequality#Isoperimetric_inequality_in_higher_dimensions)?




---


1Boáz Klartag, in a book review in the *AMS Bulletin*, July 2015, p.540.
According to the Wikipedia article, the idea goes back to Paul Lévy.



How to explain the concentration-of-measure phenomenon intuitively?
As I see it, the key intuition is passing from the equator orthogonal to a single vector to looking at a whole orthonormal basis.


Suppose we pick a random unit vector $(x\_1,\dots,x\_n)$. What we want to know is why $x\_1$ is probably near zero, since this is equivalent to being near the equator relative to the first basis vector. But this feels intuitively obvious to me: all the coordinates have the same distribution, and they surely can't all be large, so they had better all be small.


To be a little more precise, we have $x\_1^2+\dots+x\_n^2=1$, and each coordinate has the same distribution, so the expected value of $x\_1^2$ is $1/n$. Now we can just apply Markov's inequality. For example, the probability that $|x\_1|$ is at least $1/n^{1/4}$ must be at most $1/n^{1/2}$, since otherwise the expected value of $x\_1^2$ would be too large.


(This is not so different from Bjørn and Dustin's answers, but expressed in a less sophisticated way.)


