The KLS Conjecture (problem 30)

The isoperimetric problem dates back to the ancient Greeks who conjectured that in $\mathbb{R}^2$, the shape with fixed boundary length that maximizes is given by a circle. While this has been proven a long time ago, the same question for higher-dimensional spaces and general measures remains an active area of research. For a probability measure $\mu$ on $\mathbb{R}^n$, and Euclidean distance $d$, the isoperimetric problem consists in determining the set $S$ that minimizes the boundary measure $\mu^+(S)$ given a fixed volume $\mu(S)$.

Definition 1:

The boundary measure of a set $S \subset \mathbb{R}^n$ with respect to the probability measure $\mu$ and Euclidean distance $d$ is given by \(\mu^+(S):= \lim_{\varepsilon \to 0}\frac{\mu(S+\varepsilon B_n)-\mu(S)}{\varepsilon},\) where $B_n$ denotes the Euclidean Ball in $n$ dimensions.

The isoperimetric constant $\psi_\mu$ is the minimal boundary measure among sets $S$ with $\mu(S)\geq \frac{1}{2}$: \(\psi_\mu := \inf_{S \subseteq \mathbb{R}^n} \frac{\mu^+(S)}{\min\{\mu(S), \mu(\mathbb{R}^n\backslash S)\}}.\)

Sometimes $\psi_\mu$ is referred to as the Cheeger constant. For the $n$-dimensional standard Gaussian measure $\gamma_n$, the isoperimetric constant is given by the minimum over half-spaces (first proven in (Sudakov & Tsirel’son, 1974), but several proofs exist nowadays). Notably, this implies lower bounds on the isoperimetric constant independent of the dimension.

On the other hand, a general measure $\mu$ might admit a bottleneck, i.e. a split dividing $\mathbb{R}^n$ into two or more regions that are only connected by small probability sets (the reader may want to image a gym dumbbell), and in those cases the isoperimetric constant can be arbitrarily small. This motivates the following question: Does there exist a class of measures on $\mathbb{R}^n$ where we do not expect bottlenecks, or more precisely, such that the isoperimetric constant is bounded from below by a dimension-independent and universal constant?

Consider the class of log-concave probability distributions. These are the probability measures with densities $f$ such that $\log(f)$ is concave. This includes in particular all uniform measures over convex sets, a situation where we do not expect bottlenecks to appear. Indeed, Kannan, Lovász and Simonovits (Kannan et al., 1995) conjectured in 1995 that for all log-concave measures, the isoperimetric constant is, up to a universal constant, attained by minimizing over half-spaces, like in the Gaussian case:

Conjecture 30 (KLS Conjecture)

There exists a universal constant $C>0$ such that for every log-concave measure $\mu$, the isoperimetric constant satisfies \begin{equation} \psi_\mu \geq C \inf_{H \subseteq \mathbb{R}^n, H \text{ is halfspace}} \frac{\mu^+(H)}{\min\{\mu(H), \mu(\mathbb{R}^n\backslash H)\}}. \end{equation}
Equivalently, there exists a universal constant $C>0$ such that \(\psi_\mu \geq \frac{C}{\sqrt{\\||A\\||_\text{op}}},\) where $A$ denotes the covariance matrix of the measure $\mu$, $A=\mathbb{E} \left(X- \mathbb{E}X\right)\left(X- \mathbb{E}X\right)^\top$, where $\mathbb{E}$ denotes expectation of $X\sim\mu$.

The equivalence is due to log-concavity and shown in Lemma 1 in (Lee & Vempala, 2018). Moreover, by this equivalence we can focus on isotropic distributions, i.e. $\mathbb{E}[X]=0$ and $A= I$. We will denote by $\psi_n$ the infimum over all log-concave isotropic distributions over $\mathbb{R}^n$.

Besides important connections to other conjectures which we will discuss later, a lower bound on the isoperimteric constant has very useful implications:

• It is equivalent to a Poincaré inequality which states that there exists a constant $C_P>0$ such that for all smooth functions $f: \mathbb{R}^n \to \mathbb{R}$, \(\mathrm{Var}_{X \sim \mu}(f(X)) \leq C_P \int_{\mathbb{R}^n} \\|\nabla f\\|_2^2 \,d \mu.\)

• Moreover, a lower bound on the isoperimetric constant is equivalent to Lipschitz concentration (Gromov & Milman, 1983), (Milman, 2009): For $X \sim \mu$ and $f$ being an $L$-Lipschitz function, it holds \(\mathbb{P}(|f(X)-\mathbb{E}[f(X)]|\geq t) \leq \exp(-\frac{t}{L}\Omega(\psi_\mu)).\)
• The ball walk on a convex body is a Markov Chain Monte Carlo algorithm to sample from the uniform distribution over a convex body. If the starting distribution is sufficiently close to the uniform distribution, the KLS conjecture implies that for every convex body the ball walk mixes in $\tilde{O}(n^2)$ steps (up to log factors (Kannan et al., 1997)). This rate is, in general, unimprovable.

For a more detailed description and further applications, see the survey by Lee and Vempala (Lee & Vempala, 2018).

In recent years, there has been significant progress on improving the lower bound for the isoperimetric constant uniformly over log-concave distributions. The starting point in (Kannan et al., 1995) was the following: For one-dimensional log-concave distributions, it is well known (and simple) to deduce by concavity that the isoperimetric constant is uniformly bounded. This fact was used to reduce high-dimensional log-concave distributions to one-dimensional distributions through \textit{localization}. This technique was used to prove $\psi_\mu \geq \frac{C}{n^\frac{1}{2}}$ for isotropic distributions in (Kannan et al., 1995). Besides many important improvements, we briefly summarize the most influential improvements throughout the past fifteen years. Based on the fact that Gaussians and measures which are \textit{at least as} log-concave as a Gaussians, admit good isoperimetric constants (Bakry & Ledoux, 1996), it has been tried to reduce the measure $\mu$ to a more log-concave measure.

Instead of the localization approach, Ronen Eldan (Eldan, 2013) introduced the \textit{stochastic localization} approach. The idea is to change the measure $\mu$ with a stochastic process over time $t$. This yields a measure decomposition of $\mu$ into a mixture of Gaussians $\mu_t$ with isoperimetric constants $\sqrt{t}$ improving over time. The process is then stopped after a short amount of time $T$, such that $\frac{\mu_T(S)}{\mu(S)}$ is constant with high probability for sets with ${\mu(S)}=\frac{1}{2}$, i.e. it preserves constant mass in large sets. Hence, the isoperimetric constant can be bounded by $\sqrt{T}$ and a proof then breaks down to find the best time $T$ that guarantees the boundedness of $\frac{\mu_T(S)}{\mu(S)}$.

Eldan (Eldan, 2013) proved $\psi_n \geq Cn^{-\frac{1}{3}}\log(n)^{-\frac{1}{2}}$, which was later improved by Lee and Vempala (Lee & Vempala, 2017) to $\psi_n \geq Cn^{-\frac{1}{4}}$ and believed to be optimal for some time.

This changed due to a breakthrough, achieved by Yuansi Chen (Chen, 2021) proving $\psi_n \geq C\exp(-\sqrt{\log(n)\log(\log(n))})$ using stochastic localization and a bootstrapping argument, breaking the polynomial bottleneck. This was recently improved by Klartag (Klartag, 2023) using an \textit{improved Lichnerowicz inequality} to obtain $\psi_n \geq C \log(n)^{-\frac{1}{2}}$, which is currently the best known lower bound.

The KLS Conjecture can also be related to two other conjectures which have been resolved in the past months: The Thin Shell Conjecture was proven by Klartag and Lehec (Klartag & Lehec, 2025) and states the following:

Theorem 1:

For any isotropic log-concave measure $\mu$ on $\mathbb{R}^n$, the mass concentrates around a thin spherical shell of radius $\sqrt{n}$ and constant width, i.e. there exists a universal constant $C>0$ such that \(\mathbb{E}((\\|X\\|_2 - \sqrt{n})^2) \leq C .\)

The fact that the KLS conjecture implies the Thin Shell Conjecture can be readily seen by applying the Poincaré inequality. Remarkably, Eldan (Eldan, 2013) proved the reverse up to $\sqrt{\log(n)}$ using stochastic localization. In particular, the resolution of the Thin Shell Conjecture in 2025 (Klartag & Lehec, 2025) also implies the current best bound for the KLS Conjecture.

Another famous problem from high-dimensional convex geometry is Bourgain’s slicing problem (Bourgain, 1986) from 1986 (now a theorem of Klartag and Lehec (Klartag & Lehec, 2025)):

Theorem 2:

There exists a universal constant $C>0$ such that for every dimension $n$ and every convex body $K \subset \mathbb{R}^n$ with unit volume there exists a hyperplane $H \subset \mathbb{R}^n$ such that $\mathrm{Vol}_{n-1}(K \cap H) \geq C$.

Moreover, it can be reformulated (Ball, 1988) for the class of log-concave isotropic measures with density $f$: There is a universal constant $C>0$ such that $f(0)^\frac{1}{n} \leq C$. Remarkably, the slicing problem has been posed before the KLS Conjecture and the connection was not known for a long time: Eldan and Klartag (Eldan & Klartag, 2011)© used the log-Laplace transform to show that the Thin Shell Conjecture implies an affirmative answer to Bourgain’s problem and therefore the KLS Conjecture also implies Bourgain’s slicing conjecture. Based on a new technique due to Guan (Guan, 2024), the problem was first resolved in December 2024 by Klartag and Lehec (Klartag & Lehec, 2025), before the same authors proved the more general Thin Shell bound a few months later.

References

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