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Thin shell conjecture constant (C20aC_{20a})

Thin shell conjecture constant: determine C20aC_{20a} or improve its bounds; the repository records 2 (lower) and <∞< \infty (upper).

MSC 52A23 · Asymptotic theory of convex bodiesMSC 60E15 · Inequalities; stochastic orderings

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

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

We define C20aC_{20a} to be the smallest constant such that

Var(∣X∣2)  =  E(∣X∣2−n)2 ≤ C20a n\mathrm{Var}(|X|^2) \;=\; \mathbb E\bigl(|X|^2-n\bigr)^2 \ \le\ C_{20a}\,n

for every dimension nn and every isotropic log-concave XX in Rn\mathbb R^n.

Equivalently,

C20a=sup⁡n≥1 sup⁡X Var(∣X∣2)n,C_{20a}=\sup_{n\ge 1}\ \sup_{X}\ \frac{\mathrm{Var}(|X|^2)}{n},

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

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

E(∣X∣−n)2 ≤ 1n E(∣X∣2−n)2 ≤ C20a,\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 C20aC_{20a} means that ∣X∣\lvert X\rvert concentrates in a shell of constant width around n\sqrt n.

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Background

  • Thin shell implies slicing (hyperplane) bounds (see [EK2011]). Thus [KL2025] gives (independently) slicing-type control, complementing the direct slicing resolution [KL2024].
  • Many of the improvements toward thin shell proceeded via progress on the KLS isoperimetric constant; see [Che2021], [KL2022], [JLV2022], [K2023], [Gua2024].

References (from the repository page)

  • [ABP2003] Anttila, M.; Ball, K.; Perissinaki, I. The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 (2003), no. 12, 4723–4735.
  • [BK2003] Bobkov, S. G.; Koldobsky, A. On the central limit property of convex bodies. In: Geometric aspects of functional analysis (2001–02), Lecture Notes in Math. 1807, Springer (2003), 44–52.
  • [Che2021] Chen, Y. An almost constant lower bound of the isoperimetric coefficient in the KLS conjecture. Geom. Funct. Anal. 31 (2021), no. 1, 34–61.
  • [EK2011] Eldan, R.; Klartag, B. Approximately Gaussian marginals and the hyperplane conjecture. In: Concentration, functional inequalities and isoperimetry, Contemp. Math. 545, Amer. Math. Soc. (2011), 55–68.
  • [Fle2010] Fleury, B. Concentration in a thin Euclidean shell for log-concave measures. J. Funct. Anal. 259 (2010), no. 4, 832–841.
  • [GM2011] Guédon, O.; Milman, E. Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21 (2011), no. 5, 1043–1068.
  • [Gua2024] Guan, Q. A note on Bourgain’s slicing problem. Preprint (2024). arXiv:2412.09075
  • [JLV2022] Jambulapati, A.; Lee, Y. T.; Vempala, S. S. A slightly improved bound for the KLS constant. Preprint (2022). arXiv:2208.11644
  • [K2007a] Klartag, B. A central limit theorem for convex sets. Invent. Math. 168 (2007), no. 1, 91–131.
  • [K2007b] Klartag, B. Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 (2007), no. 1, 284–310.
  • [K2023] Klartag, B. Logarithmic bounds for isoperimetry and slices of convex sets. Ars Inveniendi Analytica, Paper No. 4 (2023), 17pp.
  • [KL2022] Klartag, B.; Lehec, J. Bourgain’s slicing problem and KLS isoperimetry up to polylog. Geom. Funct. Anal. 32 (2022), no. 5, 1134–1159.
  • [KL2024] Klartag, B.; Lehec, J. Affirmative resolution of Bourgain’s slicing problem using Guan’s bound. Preprint (2024). arXiv:2412.15044
  • [KL2025] Klartag, B.; Lehec, J. Thin-shell bounds via parallel coupling. Preprint (2025). arXiv:2507.15495
  • [LV2017] Lee, Y. T.; Vempala, S. Eldan’s stochastic localization and the KLS hyperplane conjecture: an improved lower bound for expansion. In: FOCS 2017, 998–1007.
  • [Pao2006] Paouris, G. Concentration of mass on convex bodies. Geom. Funct. Anal. 16 (2006), no. 5, 1021–1049.

Acknowledgements

Prepared with ChatGPT 5.2 Pro.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 2; best known upper bound <∞< \infty. Values marked with an asterisk have minimal verification.

Known upper bounds

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

σn2:=sup⁡XE(∣X∣−n)2,\sigma_n^2 := \sup_X \mathbb E\bigl(|X|-\sqrt n\bigr)^2,

where the supremum is over isotropic log-concave XX in Rn\mathbb R^n. Any bound σn≤f(n)\sigma_n \le f(n) is evidence toward (and is closely related to) boundedness of C20aC_{20a}.

Bound Reference Comments
σn≤O(nlog⁡n)\sigma_n \le O\left(\sqrt{\frac{n}{\log n}}\right) [K2007a] First nontrivial bound.
σn≤O(n2/5+o(1))\sigma_n \le O\left(n^{2/5+o(1)}\right) [K2007b] Improvement via power-law CLT methods.
σn≤O(n3/8)\sigma_n \le O(n^{3/8}) [Fle2010] Further improvement.
σn≤O(n1/3)\sigma_n \le O(n^{1/3}) [GM2011] “Thin-shell / large deviation interpolation” bound.
σn≤O(n1/4)\sigma_n \le O(n^{1/4}) [LV2017] Via Eldan’s stochastic localization.
σn≤exp⁡((log⁡n)1/2+o(1))=no(1)\sigma_n \le \exp\bigl((\log n)^{1/2+o(1)}\bigr)=n^{o(1)} [Che2021] First subpolynomial bound (via near-constant KLS).
σn≤O(log⁡4n)\sigma_n \le O(\log^4 n) [KL2022] Polylog bound.
σn≤O(log⁡2.23…n)\sigma_n \le O(\log^{2.23\ldots} n) [JLV2022] Improves the polylog exponent.
σn≤O(log⁡n)\sigma_n \le O(\sqrt{\log n}) [K2023] Further improvement.
σn≤O(log⁡log⁡n)\sigma_n \le O(\log\log n) [Gua2024] Based on a log⁡log⁡n\log\log n KLS bound.
σn≤O(1)\sigma_n \le O(1) (and in fact Var(∣X∣2)≤Cn\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
00 Trivial By definition.
4/5=0.84/5 = 0.8 [KL2025] Achieved by the cube (for the variance formulation).
22 [KL2025] Achieved by the standard Gaussian: if X∼N(0,Id)X\sim N(0,\mathrm{Id}) then Var(∣X∣2)=2n\mathrm{Var}(\lvert X\rvert^2)=2n, so C20a≥2C_{20a}\ge 2.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Thin shell conjecture constant ($C_{20a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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