There are various "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!