KLS Conjecture (Kannan–Lovász–Simonovits)

§ Problem Statement

Setup

For each integer $n \ge 1$ , let $\mu$ be a Borel probability measure on $\mathbb{R}^n$ that is absolutely continuous with respect to Lebesgue measure, with density $f:\mathbb{R}^n \to [0,\infty)$ of the form $f(x)=e^{-V(x)}$ , where $V:\mathbb{R}^n\to(-\infty,\infty]$ is convex; equivalently, $\mu$ is log-concave. Assume $\mu$ is isotropic, meaning

\int_{\mathbb{R}^n} x\,d\mu(x)=0 \quad\text{and}\quad \int_{\mathbb{R}^n} x x^\top\, d\mu(x)=I_n,

where $I_n$ is the $n\times n$ identity matrix.

For a measurable set $A\subseteq\mathbb{R}^n$ , define its $\varepsilon$ -enlargement by $A_\varepsilon=\{x\in\mathbb{R}^n:\operatorname{dist}(x,A)\le \varepsilon\}$ and its Minkowski boundary measure (with respect to $\mu$ ) by

\mu^+(A)=\liminf_{\varepsilon\downarrow 0}\frac{\mu(A_\varepsilon)-\mu(A)}{\varepsilon}.

Define the Cheeger (isoperimetric) constant of $\mu$ as

\psi_\mu=\inf_{A\ \text{measurable},\ 0<\mu(A)<1}\frac{\mu^+(A)}{\min\{\mu(A),1-\mu(A)\}}.

Unsolved Problem

(Kannan–Lovász–Simonovits conjecture; Kannan et al. (1995)) determine whether there exists a universal constant $c>0$ (independent of $n$ and of $\mu$ ) such that for every dimension $n$ and every isotropic log-concave probability measure $\mu$ on $\mathbb{R}^n$ ,

\psi_\mu \ge c.

Equivalently, if $C_n$ is the smallest number such that every isotropic log-concave $\mu$ on $\mathbb{R}^n$ satisfies

\mu^+(A)\ge \frac{1}{C_n}\min\{\mu(A),1-\mu(A)\}\quad\text{for all measurable }A\subseteq\mathbb{R}^n,

the conjecture is that $\sup_{n\ge1} C_n<\infty$ (that is, $C_n=O(1)$ as $n\to\infty$ ). Eldan (2013) introduced the stochastic localization approach. The best known general bound is currently $C_n = O(\sqrt{\log n})$ , due to Klartag (2023).

§ Discussion

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§ Significance & Implications

The KLS conjecture remains a central open problem in high-dimensional convex geometry and geometric functional analysis. It would yield a dimension-free Cheeger/Poincaré scale for isotropic log-concave measures, which in turn would sharpen conductance-based guarantees for log-concave sampling algorithms (under standard oracle-model assumptions) by removing current dimension-dependent isoperimetric losses.

Status of closely related problems has changed recently: Bourgain's slicing problem was resolved in 2025 (Klartag & Lehec (2025)), while the thin-shell conjecture has a 2025 claimed resolution currently available as a preprint (Klartag & Lehec (2025)).

§ Known Partial Results

Lee & Vempala (2017): proved $\psi_\mu \gtrsim n^{-1/4}$ (equivalently $C_n = O(n^{1/4})$ ), the first major improvement via stochastic localization.
Chen (2021): improved to an almost-constant regime $\psi_\mu \ge n^{-o(1)}$ (equivalently $C_n = n^{o(1)}$ ), not yet polylogarithmic.
Klartag & Lehec (2022): obtained a polylogarithmic bound, $C_n \le (\log n)^{O(1)}$ .
Klartag (2023): improved this to $C_n = O(\sqrt{\log n})$ , the current best known general bound.
Lee et al. (2017): The conjecture is still open in full generality (dimension-free $C_n=O(1)$ remains unproved).