# Is there a counterexample to the Thin Shell Conjecture for sub-exponential distributions?

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.