There are various "[concentration-of-measure](http://en.wikipedia.org/wiki/Concentration_of_measure)" theorems, 
the best known that due to Lévy,
which is this (informally): the volume of a sphere $S^d$ in $d$ dimensions is largely
concentrated around an $\epsilon$-tubular neighborhood of an equitorial hyperplane $H$.
Here the tubular neighborhood is the set of points within distance $\epsilon$ of $H$.
I believe there is an analogous theorem for centro-symmetric convex bodies
(although I have no reference for this).
My question is:



> 
> Is there a concentration-of-measure theorem for arbitrary convex bodies $K$,
>  something along these lines: There exists a hyperplane $H$ such that 
>  most of the volume of $K$ is concentrated
>  in a tubular neighborhood of $H$?
> 
> 
> 


Or are there convex bodies so "skewed" that they resist any such section?


Likely this is known, in which case a reference would suffice. Thanks!



Concentration of measure for arbitrary convex bodies?
There are many results, and an active research industry, along these lines. In general the Euclidean ball is the best-behaved convex body in this respect, and just how similar an arbitrary convex body is depends on how you try to make your question more precise.
Here's one very general result, a special case of what is sometimes called Borell's lemma:



> 
> Let $K \subset \mathbb{R}^n$ be a convex body with volume 1 and let $A \subset \mathbb{R}^n$ be a convex symmetric set such that $vol(K\cap A) \ge 3/4$. Then for $r > 1$,
>  $$ vol(K \setminus r A) \le \frac{3}{4} 3^{-(r+1)/2}.
> $$
> 
> 
> 


Thus in particular the volume of $K$ far from a hyperplane decays exponentially.


Much sharper results exist. I mentioned a couple recent references in a comment on Gil's answer. For much more, I suggest you start by perusing [the papers of Grigoris Paouris](http://www.math.tamu.edu/~grigoris/researchpaouris.html).


