The Kannan–Lovász–Simonovits conjecture
For a probability measure $\mu$ on $\mathbb{R}^n$, the isoperimetric (Cheeger) constant is $$\psi_\mu = \inf_{S \subseteq \mathbb{R}^n} \frac{\mu^+(S)}{\min\{\mu(S), \mu(\mathbb{R}^n \setminus S)\}},$$ where $\mu^+(S)$ is the boundary measure of $S$. Kannan, Lovász and Simonovits conjectured that for log-concave $\mu$ half-spaces are, up to a universal constant, the worst sets; Lee and Vempala show that the two forms below are equivalent. Klartag proved $\psi_\mu \ge c (\log n)^{-1/2}$ for isotropic log-concave measures on $\mathbb{R}^n$. The thin-shell conjecture and Bourgain's slicing problem, both implied by the KLS conjecture, were resolved by Klartag and Lehec.
Problem. Prove or disprove that there is a universal constant $C > 0$ such that for every log-concave measure $\mu$ the isoperimetric constant satisfies $$\psi_\mu \ge C \inf_{H \subseteq \mathbb{R}^n,\ H \text{ a half-space}} \frac{\mu^+(H)}{\min\{\mu(H), \mu(\mathbb{R}^n \setminus H)\}};$$ equivalently, that there is a universal constant $C > 0$ such that $$\psi_\mu \ge \frac{C}{\sqrt{\|A\|_{\mathrm{op}}}},$$ where $A$ is the covariance matrix of $\mu$.
Source: Kannan, Lovász and Simonovits (1995). Conjecture 30 of Randomstrasse101, in Rödder's “The KLS Conjecture” (2025); Entry 13 of arXiv:2603.29571.
Imported from github.com/MColbrook/AIM.
The statement is adapted from randomstrasse101.math.ethz.ch, whose text is available under CC BY 4.0.
Classification
- Primary
- PR — Probability
- Secondary
- FA, MG
- Keywords
- convex geometry, isoperimetry, KLS conjecture, log-concave measures, Randomstrasse101
- Literature checked
- Oct. 2, 2026 by clemens-admin
References
@misc{randomstrasse2025,
author = {Bandeira, Afonso S. and Dmitriev, Daniil and Lucca, Kevin and Nizi{\'c}-Nikolac, Petar and R{\"o}dder, Almut},
title = {{Randomstrasse101}: Open Problems of 2025},
year = {2026},
eprint = {2603.29571},
archivePrefix = {arXiv}
}
@article{kannan1995isoperimetric,
author = {Kannan, Ravi and Lov{\'a}sz, L{\'a}szl{\'o} and Simonovits, Mikl{\'o}s},
title = {Isoperimetric problems for convex bodies and a localization lemma},
journal = {Discrete \& Computational Geometry},
volume = {13},
number = {3--4},
pages = {541--559},
year = {1995},
doi = {10.1007/BF02574061}
}
@misc{lee2018kls,
author = {Lee, Yin Tat and Vempala, Santosh S.},
title = {The {K}annan--{L}ov{\'a}sz--{S}imonovits conjecture},
year = {2018},
eprint = {1807.03465},
archivePrefix = {arXiv}
}
@article{klartag2023logarithmic,
author = {Klartag, Bo'az},
title = {Logarithmic bounds for isoperimetry and slices of convex sets},
journal = {Ars Inveniendi Analytica},
year = {2023},
eprint = {2303.14938},
archivePrefix = {arXiv}
}
@misc{klartag2024slicing,
author = {Klartag, Bo'az and Lehec, Joseph},
title = {Affirmative resolution of {B}ourgain's slicing problem using {G}uan's bound},
year = {2024},
eprint = {2412.15044},
archivePrefix = {arXiv}
}
@misc{klartag2025thinshell,
author = {Klartag, Bo'az and Lehec, Joseph},
title = {Thin-shell bounds via parallel coupling},
year = {2025},
eprint = {2507.15495},
archivePrefix = {arXiv}
}
Resolution claims
The Kannan--Lovász--Simonovits (KLS) conjecture asks whether every isotropic log-concave probability measure on $\mathbb R^n$ has a Cheeger constant bounded below by a universal positive constant. We prove that $ψ_n\le C$ for a universal constant $C>0$, resolving the KLS conjecture. We also prove that $C_P(μ)\le C'$ for every isotropic log-concave probability measure $μ$ on $\mathbb R^n$, with a universal constant $C'>0$.
We prove a dimension-free Poincaré inequality for isotropic unconditional log-concave probability measures, establishing the Kannan-Lovász-Simonovits (KLS) conjecture in the unconditional setting. The proof combines a new application of Dunkl operator theory combined with an appropriate transformation of the log-concave measure. Specifically, the proof follows by a direct spectral analysis of a new Dunkl-Langevin operator adapted to the transformed measure. The …
We confirm the Kannan--Lovász--Simonovits conjecture for quadratic forms: if $X \sim μ$ is an isotropic log-concave random vector in $\mathbb{R}^n$, then for any symmetric matrix $M$ one has $$ \operatorname{Var}_{X \sim μ}(\langle MX,X\rangle) \leq 2\,\mathbb{E}_{X \sim μ}|\nabla\langle MX,X\rangle|^2. $$ As an application, we apply the above to $M=\mathbb{E}_{X \sim μ}(\langle X,θ\rangle X\otimes X)$ for $θ\in S^{n-1}$ and show that the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Partially resolved → Solution claimed | The registry's curators | An $O(1)$ Bound for the KLS Constant v1 recorded from arXiv:2610.01447v2. |
| 2026-10-09 | Open → Partially resolved | The registry's curators | A dimension-free bound for isoperimetry of unconditional log-concave measures v1 recorded from arXiv:2609.38295v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00045, "The Kannan–Lovász–Simonovits conjecture", clopen.solutions.