Math Conjectures

Kannan–Lovász–Simonovits (KLS) conjecture

OPENLandmarkConjectureProposed 1995 · Canonical special case

Canonical statement

There is a universal constant C<∞C<\infty such that for every nn, every log-concave probability measure μ\mu on Rn\mathbb R^n satisfying ∫x dμ(x)=0\int x\,d\mu(x)=0 and ∫xixj dμ(x)=δij\int x_ix_j\,d\mu(x)=\delta_{ij}, and every locally Lipschitz f:Rn→Rf:\mathbb R^n\to\mathbb R,
Var⁡μ(f)≤C∫Rn∥∇f(x)∥22 dμ(x). \operatorname{Var}_\mu(f) \le C\int_{\mathbb R^n}\|\nabla f(x)\|_2^2\,d\mu(x).
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There is a universal constant \(C<\infty\) such that for every \(n\), every log-concave probability measure \(\mu\) on \(\mathbb R^n\) satisfying \(\int x\,d\mu(x)=0\) and \(\int x_ix_j\,d\mu(x)=\delta_{ij}\), and every locally Lipschitz \(f:\mathbb R^n\to\mathbb R\), \[ \operatorname{Var}_\mu(f) \le C\int_{\mathbb R^n}\|\nabla f(x)\|_2^2\,d\mu(x). \]
ANAL-FA-015:[KannanLovaszSimonovits1995Isoperimetric][LeeVempala2018KLSsurvey][KlartagLehec2022SlicingKLS][KlartagLehec2025SlicingSolved]
Analysis and functional analysisProbability theoryMathematical optimization

Notes

The KLS conjecture of Kannan, Lovász and Simonovits (1995) asserts a dimension-free Poincaré inequality for isotropic log-concave measures: there is a universal constant CC such that every centered log-concave probability measure on Rn\mathbb R^n with identity covariance satisfies Var⁡μ(f)≤C∫∥∇f∥22 dμ\operatorname{Var}_\mu(f)\le C\int\|\nabla f\|_2^2\,d\mu for all locally Lipschitz ff. Informally, up to constants, the worst way to split such a measure into two parts is with a hyperplane cut; the conjecture arose from the authors' localization-lemma study of isoperimetric problems for convex bodies [KannanLovaszSimonovits1995Isoperimetric].

The known bounds on the KLS constant have improved dramatically through stochastic localization, a development surveyed in [LeeVempala2018KLSsurvey]. Klartag and Lehec proved a bound polylogarithmic in the dimension, simultaneously yielding polylogarithmic bounds for Bourgain's slicing problem [KlartagLehec2022SlicingKLS], and the slicing problem itself has since been resolved affirmatively [KlartagLehec2025SlicingSolved]. That resolution, however, does not by itself settle KLS.

The conjecture proper — a truly dimension-free constant — remains open; what is missing is the final step from nearly constant, polylogarithmic bounds to a universal CC independent of nn.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.