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OP-12024OpenGeometry and topology

Isotropic constant of a log-concave probability measure (C20bC_{20b})

Isotropic constant of a log-concave probability measure: determine C20bC_{20b} or improve its bounds; the repository records 1/e1/e (lower) and <∞< \infty (upper).

MSC 52A23 · Asymptotic theory of convex bodies

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 ↓
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Problem statement

Let μ\mu be a Borel probability measure on Rn\mathbb{R}^n with finite second moments. Its covariance matrix is

Cov(μ) := ∫Rn(x−m)(x−m)T dμ(x),m:=∫Rnx dμ(x).\mathrm{Cov}(\mu) \ :=\ \int_{\mathbb{R}^n} (x-m)(x-m)^{\mathsf T}\, d\mu(x), \qquad m:=\int_{\mathbb{R}^n} x\, d\mu(x).

Convex bodies

If K⊂RnK\subset\mathbb{R}^n is a convex body, let λK\lambda_{K} be the uniform probability measure on KK and abbreviate Cov(K):=Cov(λK)\mathrm{Cov}(K):=\mathrm{Cov}(\lambda_{K}). The isotropic constant of KK is

LK := (det⁡Cov(K)Voln(K)2)1/(2n).L_{K} \ :=\ \left(\frac{\det \mathrm{Cov}(K)}{\mathrm{Vol}_{n}(K)^2}\right)^{1/(2n)}.

This quantity is invariant under invertible affine maps.

Define

Lnbody := sup⁡{LK: K⊂Rn a convex body}.L_{n}^{\mathrm{body}} \ :=\ \sup\{L_{K}:\ K\subset\mathbb{R}^n \text{ a convex body}\}.

Log-concave probability measures

If μ\mu is absolutely continuous with density ff, its (differential) entropy is

Ent(μ) := −∫Rnflog⁡f.\mathrm{Ent}(\mu)\ :=\ -\int_{\mathbb{R}^n} f \log f.

For an absolutely-continuous log-concave probability measure μ\mu on Rn\mathbb{R}^n, define its isotropic constant by

Lμ := e−Ent(μ)/n⋅det⁡(Cov(μ))1/(2n).L_\mu\ :=\ e^{-\mathrm{Ent}(\mu)/n}\cdot \det(\mathrm{Cov}(\mu))^{1/(2n)}.

(If μ\mu is log-concave but supported on a proper affine subspace, define LμL_{\mu} in that subspace.)

If KK is a convex body, then λK\lambda_{K} has constant density 1/Voln(K)1/\mathrm{Vol}_{n}(K) on KK, hence Ent(λK)=log⁡Voln(K)\mathrm{Ent}(\lambda_{K})=\log\mathrm{Vol}_{n}(K), and therefore

LλK:=e−log⁡(Voln(K))/n⋅det⁡(Cov(K))1/(2n):=(det⁡Cov(K)Voln(K)2)1/(2n):=LK.L_{\lambda_{K}} := e^{-\log(\mathrm{Vol}_{n}(K))/n}\cdot \det(\mathrm{Cov}(K))^{1/(2n)} := \left(\frac{\det \mathrm{Cov}(K)}{\mathrm{Vol}_{n}(K)^2}\right)^{1/(2n)} := L_{K}.

Define

Lnlc := sup⁡{Lμ: μ log-concave on Rn},C20b := sup⁡n≥1Lnlc.L_{n}^{\mathrm{lc}} \ :=\ \sup\{L_\mu:\ \mu \text{ log-concave on }\mathbb{R}^n\}, \qquad C_{20b} \ :=\ \sup_{n\ge 1} L_{n}^{\mathrm{lc}}.

The isotropic constant problem asked whether C20b<∞C_{20b}<\infty (i.e. whether LμL_\mu is bounded by a universal constant, independent of the dimension). This is now known to be true.

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Background

  • Convex bodies vs. log-concave measures. Since λK\lambda_{K} is log-concave and LλK=LKL_{\lambda_{K}}=L_{K}, we have Lnbody≤LnlcL_{n}^{\mathrm{body}}\le L_{n}^{\mathrm{lc}}. Conversely, Ball [Ball1988] (even case) and Klartag [K2006] (general case) show that bounding isotropic constants for convex bodies yields (up to universal constants) bounds for arbitrary log-concave measures, via Ball’s associated convex bodies (“Ball bodies” / KpK_p-bodies). Hence finiteness of sup⁡nLnbody\sup_{n} L_{n}^{\mathrm{body}} and of C20b=sup⁡nLnlcC_{20b}=\sup_{n} L_{n}^{\mathrm{lc}} are equivalent up to universal factors.

  • Connection to the slicing (hyperplane) theorem. A standard formulation of Bourgain’s slicing/hyperplane problem is precisely the assertion that sup⁡nLnbody<∞\sup_{n} L_{n}^{\mathrm{body}}<\infty. In particular, finiteness of C20bC_{20b} implies the slicing/hyperplane conclusion that every convex body of volume 11 has a hyperplane section of (n−1)(n-1)-dimensional volume bounded below by a universal constant.

  • Conjectured sharp value. It is conjectured that sup⁡nLnbody=1/e\sup_{n} L_{n}^{\mathrm{body}} = 1/e (simplex extremizers), and that among centrally-symmetric bodies the cube is extremal (giving 1/121/\sqrt{12}); see [KL2024] for discussion.

References (from the repository page)

  • [Ball1988] Ball, K. Logarithmically concave functions and sections of convex sets in Rn\mathbb{R}^n. Studia Math. 88 (1988), no. 1, 69–84.
  • [Bou1991] Bourgain, J. On the distribution of polynomials on high dimensional convex sets. In: Geometric Aspects of Functional Analysis (Israel Seminar 1989–90), Lecture Notes in Math. 1469, Springer (1991), 127–137.
  • [Bou2002] Bourgain, J. On the isotropy-constant problem for “Ψ\Psi-2” bodies. In: Geometric Aspects of Functional Analysis (Israel Seminar 2001–02), Lecture Notes in Math. 1807, Springer (2002), 114–121.
  • [K2006] Klartag, B. On convex perturbations with a bounded isotropic constant. Geom. Funct. Anal. 16 (2006), 1274–1290.
  • [Che2021] Chen, Y. An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture. Geom. Funct. Anal. 31 (2021), 34–61.
  • [KL2022] Klartag, B.; Lehec, J. Bourgain’s slicing problem and KLS isoperimetry up to polylog. Geom. Funct. Anal. 32 (2022), no. 5, 1134–1159. arXiv:2203.15551
  • [JLV2022] Jambulapati, A.; Lee, Y. T.; Vempala, S. A Slightly Improved Bound for the KLS Constant. arXiv:2208.11644
  • [K2023] Klartag, B. Logarithmic bounds for isoperimetry and slices of convex sets. Ars Inveniendi Analytica, Paper No. 4 (2023), 17 pp.
  • [Gua2024] Guan, Q. A note on Bourgain’s slicing problem. arXiv:2412.09075
  • [KL2024] Klartag, B.; Lehec, J. Affirmative Resolution of Bourgain’s Slicing Problem using Guan’s Bound. arXiv:2412.15044. (Published in Geom. Funct. Anal. 35 (2025), 1147–1168.)

Acknowledgements

Prepared with ChatGPT 5.2 Pro.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1/e1/e; best known upper bound <∞< \infty. Values marked with an asterisk have minimal verification.

Known upper bounds

Below, bounds are stated for LnbodyL_{n}^{\mathrm{body}} (equivalently for LnlcL_{n}^{\mathrm{lc}} up to universal factors; see comments).

Bound Reference Comments
Lnbody≤C n1/4log⁡nL_{n}^{\mathrm{body}} \le C\,n^{1/4}\log n [Bou1991], [Bou2002] Bourgain’s classical bound
Lnbody≤C n1/4L_{n}^{\mathrm{body}} \le C\,n^{1/4} [K2006] First removal of the log⁡n\log n factor
Lnbody≤exp⁡(Clog⁡n log⁡log⁡n)L_{n}^{\mathrm{body}} \le \exp\big(C\sqrt{\log n}\,\log\log n\big) [Che2021] First “subpolynomial” bound
Lnbody≤C (log⁡n)4L_{n}^{\mathrm{body}} \le C\,(\log n)^4 [KL2022] First polylogarithmic bound
Lnbody≤C (log⁡n)2.223…L_{n}^{\mathrm{body}} \le C\,(\log n)^{2.223\ldots} [JLV2022]
Lnbody≤C (log⁡n)2.082…L_{n}^{\mathrm{body}} \le C\,(\log n)^{2.082\ldots} [K2023] Lehec (personal communication)
Lnbody≤C log⁡nL_{n}^{\mathrm{body}} \le C\,\sqrt{\log n} [K2023]
Lnbody≤C log⁡log⁡nL_{n}^{\mathrm{body}} \le C\,\log\log n [Gua2024]
C20b<∞C_{20b}<\infty (dimension-free) [KL2024] Final dimension-free bound (slicing/hyperplane theorem)

Known lower bounds

These are lower bounds for the extremal constant C20b=sup⁡nLnlcC_{20b}=\sup_{n} L_{n}^{\mathrm{lc}} (i.e. examples with large isotropic constant).

Bound Reference Comments
C20b≥1/e≈0.367879C_{20b}\ge 1/e \approx 0.367879 Classical Achieved asymptotically by simplices (their isotropic constants tend to 1/e1/e)
C20b≥1/12≈0.288675C_{20b}\ge 1/\sqrt{12}\approx 0.288675 Classical Achieved by the cube [−12,12]n[-\tfrac12,\tfrac12]^n (volume 11)

(Separately: for every log-concave probability measure μ\mu, one has the universal lower bound Lμ≥1/2πeL_\mu \ge 1/\sqrt{2\pi e}, with equality for Gaussian measures; see [KL2024].)

References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Isotropic constant of a log-concave probability measure ($C_{20b}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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