KLS Conjecture

The Kannan-Lovász-Simonovits conjecture, usually called the KLS conjecture, concerns how efficiently a convex body can be divided into two parts. Proposed by Ravi Kannan, László Lovász and Miklós Simonovits in 1995, it predicts that a hyperplane cut comes within a universal constant factor of the best possible cut.^[1]

October 2026 preprints by Bizeul, Klartag and Lehec, and by Song and Zhang, claimed proofs. The former reports substantial ChatGPT assistance. Independent validation is a separate question.^[4][5]

Geometric statement

For a probability density, the Cheeger constant compares boundary measure with the probability of the smaller of the two separated regions. It takes the infimum of this ratio over admissible regions. KLS predicts that restricting those regions to half-spaces changes the answer by at most a constant independent of dimension. It does not assert that a hyperplane is always the exact minimizing cut.^[2]

The original paper treats uniform distributions on convex bodies. Its conjecture compares isoperimetry with the largest eigenvalue of the body's covariance matrix, and explicitly relates this to nearly optimal hyperplane cuts.^[1]

Log-concavity and normalization

The modern statement covers log-concave probability distributions. A density is log-concave when its support is convex and its logarithm is concave there. Uniform distributions on convex bodies and Gaussian distributions are examples. A distribution is isotropic when its mean is zero and its covariance matrix is the identity. This fixes its location and scale before comparing different dimensions.^[6]

Poincaré formulation

The Poincaré constant $C_P(\mu)$ is the smallest constant satisfying

$$\operatorname{Var}_{\mu}(f) \leq C_P(\mu)\int \|\nabla f\|^2\,d\mu$$

for sufficiently regular functions $f$. The left side measures variation in the function's values; the right side measures the squared size of its gradient. A large constant can reflect a bottleneck in the underlying distribution.^[3]

An equivalent KLS formulation asks for a universal $C$ such that

$$\|\operatorname{Cov}(\mu)\|_{\mathrm{op}}\leq C_P(\mu)\leq C\|\operatorname{Cov}(\mu)\|_{\mathrm{op}}$$

for every log-concave probability measure. The operator norm here is the largest covariance eigenvalue. The lower bound follows by testing linear functions; the proposed upper bound is the substantive claim. For isotropic distributions, it becomes $C_P(\mu)\leq C$, independent of dimension. The Cheeger and Poincaré formulations are equivalent under log-concavity, although their constants are not identical.^[3]

For a standard Gaussian distribution in any dimension, $C_P=1$. This is one example consistent with the conjecture, not a proof for arbitrary log-concave distributions.^[3]

Why it matters for algorithms

Sampling algorithms helped motivate the conjecture. A Markov chain, such as a random walk inside a convex body, must cross between different parts of its domain to approach the target distribution. Isoperimetric bounds help control this convergence. Such sampling procedures connect to volume estimation, integration and convex optimization.^[2]

Progress before the October 2026 claims

Klartag and Lehec's 2022 work established bounds with a polylogarithmic dependence on dimension. This narrowed the gap to a dimension-independent bound without removing it.^[7]

Klartag's 2023 paper further bounded the reciprocal Cheeger constant by $C\sqrt{\log n}$ for isotropic log-concave measures. Together with the comparison between Cheeger and Poincaré constants, this gives a Poincaré bound of order $\log n$. The square root matters: a bound on the reciprocal Cheeger constant is not numerically the same bound on $C_P$.^[6]

October 2026 preprints and AI assistance

Bizeul, Klartag and Lehec

Their 32-page preprint was submitted on October 4, 2026. Its main theorem claims a dimension-independent Poincaré bound for isotropic log-concave measures.^[4]

Their argument uses stochastic localization and cumulant bounds. A suspension construction adds a coordinate in a higher-dimensional distribution, allowing control of derivatives of tilted averages. The authors then apply a criterion introduced by Song and Zhang.^[4]

The authors attribute most proofs and mathematical ideas to ChatGPT, while identifying suspension as their own contribution. They describe their work as understanding the arguments and improving their exposition.^[4]

Song and Zhang

Song and Zhang first submitted their preprint on October 1, 2026. The October 4 revision, titled An O(1) Bound for the KLS Constant, is 140 pages and claims universal bounds for both the reciprocal Cheeger constant and the Poincaré constant. The paper uses $\psi_n$ for a worst-case reciprocal Cheeger quantity: an upper bound on that quantity corresponds to a lower bound on the Cheeger constant.^[5]

Publication status

On October 12, 2026, the arXiv records showed Bizeul, Klartag and Lehec's version 1 and Song and Zhang's version 2. Neither listed a journal reference. Preprint records do not establish independent validation.^[4][5]

References

1. ^1 ^2Kannan, R., Lovász, L., and Simonovits, M. (1995). Isoperimetric Problems for Convex Bodies and a Localization Lemma. *Discrete & Computational Geometry*, 13, 541-559.
2. ^1 ^2Lee, Y. T., and Vempala, S. S. (2018). The Kannan-Lovász-Simonovits Conjecture. Survey.
3. ^1 ^2 ^3Klartag, B., and Lehec, J. (2024). Isoperimetric inequalities in high-dimensional convex sets. Lecture notes.
4. ^1 ^2 ^3 ^4 ^5Bizeul, P., Klartag, B., and Lehec, J. (2026). Presenting a proof of the Kannan-Lovasz-Simonovits conjecture. Preprint, version 1.
5. ^1 ^2 ^3Song, Z., and Zhang, X. (2026). An O(1) Bound for the KLS Constant. Preprint, version 2.
6. ^1 ^2Klartag, B. (2023). Logarithmic bounds for isoperimetry and slices of convex sets. *Ars Inveniendi Analytica*, Paper No. 4.
7. ^Klartag, B., and Lehec, J. (2022). Bourgain's slicing problem and KLS isoperimetry up to polylog.