The isotropic constant of a log-concave probability measure
Description of constant
Let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ with finite second moments. Its covariance matrix is
\[\mathrm{Cov}(\mu) \ :=\ \int_{\mathbb{R}^n} (x-m)(x-m)^{\mathsf T}\, d\mu(x), \qquad m:=\int_{\mathbb{R}^n} x\, d\mu(x).\]Convex bodies
If $K\subset\mathbb{R}^n$ is a convex body, let $\lambda_{K}$ be the uniform probability measure on $K$ and abbreviate $\mathrm{Cov}(K):=\mathrm{Cov}(\lambda_{K})$. The isotropic constant of $K$ is
\(L_{K} \ :=\ \left(\frac{\det \mathrm{Cov}(K)}{\mathrm{Vol}_{n}(K)^2}\right)^{1/(2n)}.\) This quantity is invariant under invertible affine maps.
Define
\[L_{n}^{\mathrm{body}} \ :=\ \sup\{L_{K}:\ K\subset\mathbb{R}^n \text{ a convex body}\}.\]Log-concave probability measures
If $\mu$ is absolutely continuous with density $f$, its (differential) entropy is
\[\mathrm{Ent}(\mu)\ :=\ -\int_{\mathbb{R}^n} f \log f.\]For an absolutely-continuous log-concave probability measure $\mu$ on $\mathbb{R}^n$, define its isotropic constant by
\[L_\mu\ :=\ e^{-\mathrm{Ent}(\mu)/n}\cdot \det(\mathrm{Cov}(\mu))^{1/(2n)}.\](If $\mu$ is log-concave but supported on a proper affine subspace, define $L_{\mu}$ in that subspace.)
If $K$ is a convex body, then $\lambda_{K}$ has constant density $1/\mathrm{Vol}_{n}(K)$ on $K$, hence $\mathrm{Ent}(\lambda_{K})=\log\mathrm{Vol}_{n}(K)$, and therefore
\[L_{\lambda_{K}} := e^{-\log(\mathrm{Vol}_{n}(K))/n}\cdot \det(\mathrm{Cov}(K))^{1/(2n)} := \left(\frac{\det \mathrm{Cov}(K)}{\mathrm{Vol}_{n}(K)^2}\right)^{1/(2n)} := L_{K}.\]Define
\[L_{n}^{\mathrm{lc}} \ :=\ \sup\{L_\mu:\ \mu \text{ log-concave on }\mathbb{R}^n\}, \qquad C_{20b} \ :=\ \sup_{n\ge 1} L_{n}^{\mathrm{lc}}.\]The isotropic constant problem asked whether $C_{20b}<\infty$ (i.e. whether $L_\mu$ is bounded by a universal constant, independent of the dimension). This is now known to be true.
Known upper bounds
Below, bounds are stated for $L_{n}^{\mathrm{body}}$ (equivalently for $L_{n}^{\mathrm{lc}}$ up to universal factors; see comments).
| Bound | Reference | Comments |
|---|---|---|
| $L_{n}^{\mathrm{body}} \le C\,n^{1/4}\log n$ | [Bou1991], [Bou2002] | Bourgain’s classical bound |
| $L_{n}^{\mathrm{body}} \le C\,n^{1/4}$ | [K2006] | First removal of the $\log n$ factor |
| $L_{n}^{\mathrm{body}} \le \exp\big(C\sqrt{\log n}\,\log\log n\big)$ | [Che2021] | First “subpolynomial” bound |
| $L_{n}^{\mathrm{body}} \le C\,(\log n)^4$ | [KL2022] | First polylogarithmic bound |
| $L_{n}^{\mathrm{body}} \le C\,(\log n)^{2.223\ldots}$ | [JLV2022] | |
| $L_{n}^{\mathrm{body}} \le C\,(\log n)^{2.082\ldots}$ | [K2023] | Lehec (personal communication) |
| $L_{n}^{\mathrm{body}} \le C\,\sqrt{\log n}$ | [K2023] | |
| $L_{n}^{\mathrm{body}} \le C\,\log\log n$ | [Gua2024] | |
| $C_{20b}<\infty$ (dimension-free) | [KL2024] | Final dimension-free bound (slicing/hyperplane theorem) |
Known lower bounds
These are lower bounds for the extremal constant $C_{20b}=\sup_{n} L_{n}^{\mathrm{lc}}$ (i.e. examples with large isotropic constant).
| Bound | Reference | Comments |
|---|---|---|
| $C_{20b}\ge 1/e \approx 0.367879$ | Classical | Achieved asymptotically by simplices (their isotropic constants tend to $1/e$) |
| $C_{20b}\ge 1/\sqrt{12}\approx 0.288675$ | Classical | Achieved by the cube $[-\tfrac12,\tfrac12]^n$ (volume $1$) |
(Separately: for every log-concave probability measure $\mu$, one has the universal lower bound $L_\mu \ge 1/\sqrt{2\pi e}$, with equality for Gaussian measures; see [KL2024].)
Additional comments and links
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Convex bodies vs. log-concave measures. Since $\lambda_{K}$ is log-concave and $L_{\lambda_{K}}=L_{K}$, we have $L_{n}^{\mathrm{body}}\le L_{n}^{\mathrm{lc}}$. 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” / $K_p$-bodies). Hence finiteness of $\sup_{n} L_{n}^{\mathrm{body}}$ and of $C_{20b}=\sup_{n} L_{n}^{\mathrm{lc}}$ are equivalent up to universal factors.
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Connection to the slicing (hyperplane) theorem. A standard formulation of Bourgain’s slicing/hyperplane problem is precisely the assertion that $\sup_{n} L_{n}^{\mathrm{body}}<\infty$. In particular, finiteness of $C_{20b}$ implies the slicing/hyperplane conclusion that every convex body of volume $1$ has a hyperplane section of $(n-1)$-dimensional volume bounded below by a universal constant.
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Conjectured sharp value. It is conjectured that $\sup_{n} L_{n}^{\mathrm{body}} = 1/e$ (simplex extremizers), and that among centrally-symmetric bodies the cube is extremal (giving $1/\sqrt{12}$); see [KL2024] for discussion.
References
- [Ball1988] Ball, K. Logarithmically concave functions and sections of convex sets in $\mathbb{R}^n$. 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 “$\Psi$-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.