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on Thin shell conjecture constant ($C_{20a}$)

The statement

Let $X$ be a random vector in $\mathbb R^n$ with an isotropic log-concave distribution (i.e. $X$ has a log-concave density, $\mathbb E X=0$ , and $\mathrm{Cov}(X)=\mathrm{Id}$ ). Since $X$ is isotropic, $\mathbb E|X|^2 = n$ .

We define $C_{20a}$ to be the smallest constant such that

\mathrm{Var}(|X|^2) \;=\; \mathbb E\bigl(|X|^2-n\bigr)^2 \ \le\ C_{20a}\,n

for every dimension $n$ and every isotropic log-concave $X$ in $\mathbb R^n$ .

Equivalently,

C_{20a}=\sup_{n\ge 1}\ \sup_{X}\ \frac{\mathrm{Var}(|X|^2)}{n},

where the inner supremum is over isotropic log-concave $X$ in $\mathbb R^n$ .

This “variance” formulation implies the more common thin-shell estimate

\mathbb E\bigl(|X|-\sqrt{n}\bigr)^2 \ \le\ \frac{1}{n}\,\mathbb E\bigl(|X|^2-n\bigr)^2 \ \le\ C_{20a},

so boundedness of $C_{20a}$ means that $\lvert X\rvert$ concentrates in a shell of constant width around $\sqrt n$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.