The thin shell conjecture (variance of $|X|^2$)

Description of constant

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 $\sqrt n$.

Known upper bounds

Historically, results were often phrased in terms of the (dimension-dependent) thin-shell width

\[\sigma_n^2 := \sup_X \mathbb E\bigl(|X|-\sqrt n\bigr)^2,\]

where the supremum is over isotropic log-concave $X$ in $\mathbb R^n$. Any bound $\sigma_n \le f(n)$ is evidence toward (and is closely related to) boundedness of $C_{20a}$.

Bound Reference Comments
$\sigma_n \le O\left(\sqrt{\frac{n}{\log n}}\right)$ [K2007a] First nontrivial bound.
$\sigma_n \le O\left(n^{2/5+o(1)}\right)$ [K2007b] Improvement via power-law CLT methods.
$\sigma_n \le O(n^{3/8})$ [Fle2010] Further improvement.
$\sigma_n \le O(n^{1/3})$ [GM2011] “Thin-shell / large deviation interpolation” bound.
$\sigma_n \le O(n^{1/4})$ [LV2017] Via Eldan’s stochastic localization.
$\sigma_n \le \exp\bigl((\log n)^{1/2+o(1)}\bigr)=n^{o(1)}$ [Che2021] First subpolynomial bound (via near-constant KLS).
$\sigma_n \le O(\log^4 n)$ [KL2022] Polylog bound.
$\sigma_n \le O(\log^{2.23\ldots} n)$ [JLV2022] Improves the polylog exponent.
$\sigma_n \le O(\sqrt{\log n})$ [K2023] Further improvement.
$\sigma_n \le O(\log\log n)$ [Gua2024] Based on a $\log\log n$ KLS bound.
$\sigma_n \le O(1)$ (and in fact $\mathrm{Var}(\lvert X\rvert^2)\le C n$) [KL2025] Affirmative resolution of the thin shell conjecture. The universal constant is not optimized (and is not made explicit).

Known lower bounds

Bound Reference Comments
$0$ Trivial By definition.
$4/5 = 0.8$ [KL2025] Achieved by the cube (for the variance formulation).
$2$ [KL2025] Achieved by the standard Gaussian: if $X\sim N(0,\mathrm{Id})$ then $\mathrm{Var}(\lvert X\rvert^2)=2n$, so $C_{20a}\ge 2$.

References

Acknowledgements

Prepared with ChatGPT 5.2 Pro.