The thin shell conjecture states that there exist universal constants $C,c>0$ such that every logconcave isotropic random vector $X$ in every Euclidean space $\mathbb{R}^n$ satisfies

$$\mathbb{P}\Big\{\big|\|X\|_2-\sqrt{n}\big|\geq t\Big\}\leq Ce^{-ct}\qquad \forall t\geq0.$$

That is, most of the probability mass resides in a constant-thickened sphere of radius $\sqrt{n}$, i.e, a "thin shell." A random vector $X$ in $\mathbb{R}^n$ is said to be sub-exponential if there exist constants $c_1,c_2>0$ such that

$$\mathbb{P}\{|\langle X,v\rangle|\geq t\}\leq c_1e^{-c_2t} \qquad \forall v\in S^{n-1}, ~t\geq0.$$

One may verify that every logconcave isotropic distribution is sub-exponential with $c_1=1$ and $c_2$ being the reciprocal of Paouris' constant.

Question: Does there exist a sequence of random vectors of increasing dimension that are isotropic and sub-exponential with fixed constants $c_1,c_2>0$, but fail to reside in thin shells?

While I'm having difficulty producing such a counterexample, I never read about the thin shell conjecture beyond logconcave distributions, so perhaps a counterexample is known. Or maybe any sub-exponential counterexample can be transformed into a logconcave counterexample?


Ahh, just take $X$ to be $0$ with probability $1/2$ and otherwise draw uniformly from the sphere of radius $\sqrt{2n}$.

If the support of the distribution were convex, then it wouldn't be able to "skip" the thin shell. I guess this is why logconcavity is important.


Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.