Convex geometry · log-concave probability · stochastic localization

Kannan–Lovász–Simonovits Conjecture

Collaboration beta

Does every isotropic log-concave probability distribution have a dimension-independent lower bound on how much boundary is needed to cut off a given share of its mass?

μ^{+} (E) = {liminf}_{ϵ ↓ 0} \frac{μ (E + ϵ B_{2}^{n}) - μ (E)}{ϵ} .

Listed inRandomstrasse 101 Open Problems 2025 — Problem 30

Known results and sources

KLS asks whether every isotropic log-concave distribution pays a universal amount of boundary when its mass is divided.

Research problem

Exact mathematical statement

Let $\mu$ be a log-concave probability measure on $\mathbb R^n$ with mean $0$ and covariance $I_n$ . For measurable $E\subseteq\mathbb R^n$ , define its outer Minkowski boundary measure by

\mu^+(E)=\liminf_{\varepsilon\downarrow0}\frac{\mu(E+\varepsilon B_2^n)-\mu(E)}{\varepsilon}.

Its Cheeger constant is

h(\mu)=\inf_{0<\mu(E)<1}\frac{\mu^+(E)}{\min\{\mu(E),1-\mu(E)\}}.

The Kannan–Lovász–Simonovits conjecture asks whether there is a universal constant $c>0$ , independent of $n$ and $\mu$ , such that

h(\mu)\ge c

for every isotropic log-concave probability measure $\mu$ . the source uses the source-stated central-profile reduction, under which it is enough to prove a universal lower bound for $\mu^+(E)$ whenever $\mu(E)=1/2$ .

Problem infographic

Problem at a glance

A problem-first scientific diagram defines an isotropic log-concave measure, shows a half-mass cut with its outer Minkowski boundary, gives the Cheeger ratio and universal lower-bound question, and marks the source packet's covariance-relay route as unfinished rather than proved. — The geometric question is a dimension-free lower bound on boundary expansion; the packet's stochastic-localization reductions end at an open covariance-relay exclusion.

Current mathematical picture

Where work on Kannan–Lovász–Simonovits Conjecture stands

Open conjecture

Selected route highlights from the current work. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The current work rules out several shortcuts in their stated forms: one-bit determinant information misses the exact free shear; affine-regression energy misses pure drag; fixed deterministic directions miss a path-selected terminal direction; a signed occupation compensator loses positivity; and the balanced pure-shear equations cannot be applied verbatim to the nonbalanced probe. Prove a dimension-free exclusion of a terminal-adaptive, nearly rank-one, nearly orthogonal covariance relay with vanishing information and innovation cost, after deriving the weighted nonbalanced zero-cost normal form or showing that the probe's fixed channel must pay a controlled cost.

Route status · Narrowed route

Main reductionCurrent reduction

Using the source-stated half-mass profile reduction, the current work runs mass-protecting stochastic localization: small perimeter forces large occupation entropy, then a terminal covariance spike with small occupation defect, backward locking produces an ultra-early spike, and a short auxiliary probe transfers it into an almost orthogonal nearly rank-one direction. The last relay-exclusion implication remains open.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeDerive the weighted nonbalanced zero-cost normal form for central masses with both tangent score columns and the fixed probe channel.Task status · Work already reported in progress

Work mapped so far

Kannan–Lovász–Simonovits Conjecture in numbers

1.8kretained lines of mathematical investigation1,796 in the current working snapshot

Argument development: 1,558 · 87%
Explored or eliminated routes: 52 · 3%
Open obligations: 92 · 5%
Definitions and setup: 94 · 5%

9selected mapped statements1routes investigated3open questions2contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byRavi Kannan, László Lovász, Miklós SimonovitsProposed1995Status checkedAug 7, 2026

Source recordedAug 7, 2026

Recommended next task

Prove a dimension-free persistence or maximal inequality that controls a terminal-adaptive relay direction rather than only fixed directions.

Suggested move: Use the Section 36 block variables and Schur resource law together with the near-rank-one occupation-time condition to bound the probability of large terminal selected covariance almost orthogonal to the input direction.

What would count as progress

Retain an exact proof or counterexample for the stated subproblem.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove a dimension-free persistence or maximal inequality that controls a terminal-adaptive relay direction rather than only fixed directions.Suggested move: Use the Section 36 block variables and Schur resource law together with the near-rank-one occupation-time condition to bound the probability of large terminal selected covariance almost orthogonal to the input direction.

Ready to work on

Close the analytic wrapper and process-consistency gaps after the relay mechanism is excluded.Suggested move: Port the burst and probe argument to one nonsingular signal-gated process and justify projection through zero signal, eigenvalue ties, bounded optional stopping, density and boundary smoothing, and semidefinite Brascamp–Lieb limits without changing the branchwise perimeter budget.

Ready to work on

Derive the weighted nonbalanced zero-cost normal form for central masses with both tangent score columns and the fixed probe channel.Suggested move: Starting from the arbitrary-mass covariance and signal formulas in Section 37 and Appendix A, complete the square for each channel and determine whether the fixed channel necessarily pays binary information, affine-regression innovation, or one-bit determinant drift.

Work already reported in progress

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026

Current statusOpen conjecture

Open in full generality: no dimension-free universal KLS constant is known for all isotropic log-concave measures. The best general peer-reviewed lower bound identified is of order 1/sqrt(log n). The 2025/2026 proof of the thin-shell conjecture does not remove this sqrt(log n) loss, and the maintained ETH open-problems collection continues to list KLS as open.

[5][6][7]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2025; revised 2026
PreprintKlartag and Lehec proved the thin-shell conjecture by parallel coupling. Via the known thin-shell-to-spectral-gap implication this yields the existing sqrt(log n)-loss scale, not a dimension-free KLS bound.[6]
2023
Peer reviewedKlartag sharpened the general isotropic Cheeger lower bound to order 1/sqrt(log n), the best peer-reviewed dimension dependence identified through the collection date.[5]
2022
Peer reviewedKlartag and Lehec proved the KLS lower bound up to a polylogarithmic loss and obtained a dimension-free slicing bound, a major advance that did not yet eliminate the KLS logarithms.[4]
2021
Peer reviewedChen broke the polynomial barrier by proving an n^(-o(1)) lower bound for the isoperimetric coefficient of isotropic log-concave measures.[3]

Mathematical neighborhood

Related results and reusable starting points

Current focusKannan–Lovász–Simonovits conjecture

Equivalent formulationDimension-free Poincaré inequality for isotropic log-concave measures

For log-concave measures, the Cheeger/isoperimetric and Poincaré or spectral-gap formulations are equivalent up to universal numerical constants. A dimension-free bound in either formulation is the KLS target.

[2][5]

Logical consequenceThin-shell conjecture

KLS implies dimension-free thin-shell concentration. The thin-shell conjecture is now proved in the cited preprint, but the known reverse route loses sqrt(log n), so it does not settle KLS.

[2][6]

Logical consequenceBourgain slicing conjecture

KLS implies Bourgain's slicing conjecture. Slicing has a dimension-free solution, but that conclusion is weaker and does not provide the reverse dimension-free isoperimetric estimate.

[2][4]

Related problemMixing of ball walk and log-concave sampling

A dimension-free KLS constant yields strong conductance and mixing bounds for geometric random walks used to sample log-concave distributions and estimate convex-body volume.

[1][2]

Weaker or relaxed formDimension-dependent approximate KLS bounds

Successive n^(-o(1)), inverse-polylogarithmic, and 1/sqrt(log n) lower bounds approximate the conjecture quantitatively while retaining dimension dependence.

[3][4]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA formal theory of log-concave probability measures on finite-dimensional real vector spaces, including densities, covariance, isotropic normalization, and affine transformations.
Formalization targetFormal Cheeger boundary measure and isoperimetric constants for continuous measures, with normalization-safe equivalence to Poincaré and spectral-gap inequalities.
Formalization targetFormal convex localization and stochastic localization, including evolving covariance operators, martingales, and matrix inequalities.
Formalization targetFormal thin-shell and slicing parameters plus the exact implication constants connecting them to KLS.
Formalization targetFormal conductance-to-mixing results for geometric random walks under log-concavity assumptions.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction1 of 91
lemma7 of 97

Selected mathematical clusters3 mathematical clusters

Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows

retained route statementDoes every isotropic log-concave probability distribution have a dimension-independent lower bound on how much boundary is needed to cut off a given share of its mass?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementHalf-mass reductionintermediate
retained route statementUltra-early relay reductionintermediate
retained route statementBranchwise perimeter budgetintermediate
retained route statementProbe mass controlintermediate
retained route statementFixed-direction shear boundsintermediate
retained route statementBalanced/probe scope boundaryintermediate
Recorded relationshipThe source material reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed

Open questionsSpecific obligations that remain open in the current routes.3 displayed rows

Research targetDerive the weighted nonbalanced zero-cost normal form for central masses with both tangent score columns and the fixed probe channel.in progress reported
Research targetProve a dimension-free persistence or maximal inequality that controls a terminal-adaptive relay direction rather than only fixed directions.open
Research targetClose the analytic wrapper and process-consistency gaps after the relay mechanism is excluded.open

Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.2 displayed rows · 1 route included

Useful failureSource-reported limitationreported failure
Narrowed routeSource-reported limitationThe current work rules out several shortcuts in their stated forms: one-bit determinant information misses the exact free shear; affine-regression energy misses pure drag; fixed deterministic directions miss a path-selected terminal direction; a signed occupation compensator loses positivity; and the balanced pure-shear equations cannot be applied verbatim to the nonbalanced probe. Prove a dimension-free exclusion of a terminal-adaptive, nearly rank-one, nearly orthogonal covariance relay with vanishing information and innovation cost, after deriving the weighted nonbalanced zero-cost normal form or showing that the probe's fixed channel must pay a controlled cost.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeDerive the weighted nonbalanced zero-cost normal form for central masses with both tangent score columns and the fixed probe channel.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve a dimension-free persistence or maximal inequality that controls a terminal-adaptive relay direction rather than only fixed directions.

Kannan–Lovász–Simonovits Conjecture · ready to start

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Research contextPrepared context for any AI agent

Does every isotropic log-concave probability distribution have a dimension-independent lower bound on how much boundary is needed to cut off a given share of its mass?

Exact question and boundaries
Current routes and known obstacles
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Sources and references9 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Isoperimetric Problems for Convex Bodies and a Localization Lemmaoriginal source · Ravi Kannan, László Lovász, Miklós Simonovits · Discrete & Computational Geometry · 1995 · DOI 10.1007/BF02574061 · accessed Aug 7, 2026
2
The Kannan–Lovász–Simonovits Conjecturesurvey or monograph · Yin Tat Lee, Santosh S. Vempala · 2018 · ARXIV 1807.03465 · accessed Aug 7, 2026
3
An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecturepeer reviewed result · Yuansi Chen · Geometric and Functional Analysis · 2021 · DOI 10.1007/s00039-021-00558-4 · accessed Aug 7, 2026
4
Bourgain's slicing problem and KLS isoperimetry up to polylogpeer reviewed result · Bo'az Klartag, Joseph Lehec · Geometric and Functional Analysis · 2022 · DOI 10.1007/s00039-022-00612-9 · accessed Aug 7, 2026
5
Logarithmic bounds for isoperimetry and slices of convex setspeer reviewed result · Bo'az Klartag · Ars Inveniendi Analytica · 2023 · DOI 10.15781/jsjy-0b06 · accessed Aug 7, 2026
6
Thin-shell bounds via parallel couplingpreprint · Bo'az Klartag, Joseph Lehec · 2025; revised 2026-02-23 · ARXIV 2507.15495 · accessed Aug 7, 2026
7
The KLS Conjecture (Problem 30)maintained problem list · ETH Zürich · 2025 · accessed Aug 7, 2026
8
Formal Conjectures repository tree at c594af4ba42f58465253b8550545e0132959a78cformalization · The Formal Conjectures Authors · Google DeepMind · accessed Aug 7, 2026
9
Mathlib public documentation indexformalization · The Mathlib Contributors · Mathlib · accessed Aug 7, 2026

Important qualifications

The record uses the lower-bound convention for the Cheeger constant of an isotropic log-concave measure. Sources using reciprocal constants or differing normalizations are compared only up to universal factors.
The 2025 thin-shell result remained a preprint in the sources checked, with a February 2026 revision. It proves the thin-shell conjecture but recovers KLS only with a sqrt(log n) loss and therefore is not recorded as a KLS solution.
The milestone list emphasizes best dimension dependence and major implications; it does not catalogue every symmetric-body case, covariance refinement, or sampling-algorithm consequence.
KLS implies both thin-shell and slicing statements, but their affirmative resolution does not reverse with dimension-free constants currently sufficient to prove KLS.
The scoped formalization search checked the exact current Formal Conjectures public tree and current Mathlib documentation. No problem-level KLS statement was found, but this does not establish absence from all proof assistants, branches, or private developments.
No canonical finite computation, certificate, software package, or dataset resolves or materially verifies the universal analytic statement; the principal progress consists of proofs of quantitative bounds.
No unreviewed source material, contributor claim, packet computation, or unpublished submission was inspected or used as external authority. This record grants no proof, novelty, review, acceptance, or publication authority.

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