Kannan–Lovász–Simonovits conjecture

About 19 years old · traced to 2007

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ . Write $\operatorname{Cov}(\mu)$ for its covariance matrix, $\|\operatorname{Cov}(\mu)\|_{op}$ for its operator norm, and $C_P(\mu)$ for its Poincaré constant.

Kannan–Lovász–Simonovits conjecture. There is a universal constant $C>0$ such that

\|\operatorname{Cov}(\mu)\|_{op} \leq C_P(\mu) \leq C\,\|\operatorname{Cov}(\mu)\|_{op}.

This conjecture asks whether the Poincaré constant of every log-concave measure is controlled, up to universal factors, by the largest variance of a linear function. It is a central problem concerning bottlenecks and functional inequalities in high-dimensional convexity; its resolution status is not specified in the source.

References

Primary source

Bo'az Klartag and Joseph Lehec, “Isoperimetric inequalities in high-dimensional convex sets”, arXiv:2406.01324 (2024).

Additional references

7 papers in this index state this conjecture (2007–2024). The statement above is taken from the most recent of them; the others are arXiv:2306.12997, arXiv:1810.08369, arXiv:1203.0893, arXiv:1003.4839, arXiv:0801.4036, arXiv:0712.4092.

Posted 1 month ago · edited 6 days ago

Progress summary

Refreshed 6 days ago

Claimed progress

A new preprint sharply improves the best dimension-dependent estimate, but the conjectured dimension-free bound remains unproved.

The conjecture, formulated by Kannan, Lovász, and Simonovits in 1995, predicts that for every log-concave probability measure, the Poincaré constant is controlled within a universal factor by the largest variance of a linear function. It remains open in full generality.

Known results

Kannan, Lovász, and Simonovits (1995): initial bound of order $n^{1/2}$ .
Eldan (2013): improved the bound to roughly logarithmic correction times a thin-shell parameter.
Lee and Vempala (2017): obtained an $n^{1/4}$ bound.
Chen (2021), followed by Klartag and Lehec (2022): obtained near-constant and polylogarithmic bounds, respectively, without proving a universal constant.

October 2026 iterated-logarithm bound

Zhao Song and Xinzhi Zhang’s preprint reports an improved KLS estimate of order $O(4^{\log^* n})$ , with a corresponding Poincaré bound. This is substantial quantitative progress toward dimension-independence, but it does not establish the conjecture; the preprint’s claim is not independently verified here.

Current status (as of October 2026): The conjecture remains open; substantial dimension-dependent bounds are known, including the reported $O(4^{\log^* n})$ estimate, but no universal constant bound has been established.

Sources

No solutions have been posted yet.