| { |
| "schema_version": "2.0", |
| "artifact_type": "evidence_research_draft", |
| "seed_id": "solveall_kls_conjecture", |
| "problem_input_sha256": "9ceca73deff1c904849e2e13a281ee9a443bb3cb62ea6a93b1e41e3165aa0924", |
| "researcher_model": "gpt-5.6-luna", |
| "prompt_version": "solveall-evidence-research-v1", |
| "created_at": "2026-07-27T04:15:31.011048+00:00", |
| "research_notes": "Inspected the registered sources through primary arXiv, Springer, and Annals pages. Found directly inspectable statements for the conjecture’s formulation, Chen’s almost-constant lower bound, Lee–Vempala’s n^{1/4}-scale bound, Klartag’s sqrt(log n)-factor result, and the 2025 thin-shell resolution. The 2025 thin-shell paper explicitly describes thin-shell as historically easier than KLS, so its universal thin-shell bound should not be treated as a KLS resolution. I did not add sources. The KLS 1995 original paper was not directly inspected, so no evidence card is supplied for its conjecture statement.", |
| "new_sources": [], |
| "evidence_cards": [ |
| { |
| "evidence_id": "chen-kls-conjecture-formulation", |
| "reference_statement": "For a log-concave density p in R^d, Chen states the KLS conjecture as a universal lower bound for the isoperimetric coefficient in terms of the spectral norm of the covariance; in the isotropic case this becomes a dimension-independent universal lower bound.", |
| "scope_conditions": [ |
| "p is a log-concave density on R^d", |
| "rho(p) is the spectral norm of the covariance matrix", |
| "The isotropic specialization assumes mean zero and identity covariance" |
| ], |
| "passages": [ |
| { |
| "source_id": "chen-2021", |
| "locator": "Section 1, Conjecture 1, lines 83-92 of the Springer full text", |
| "text": "There exists a universal constant c, such that for any log-concave density p in $\\mathbb {R}^d$, we have\n\n$$\\begin{aligned} \\psi (p) \\ge \\frac{c}{\\sqrt{\\rho \\left( p \\right) }}, \\end{aligned}$$\n\nwhere $\\rho \\left( p \\right) $ is the spectral norm of the covariance matrix of p. In other words, $\\rho \\left( p \\right) = \\left\\| A\\right\\| _{2}$, where $A = {{\\,\\mathrm{Cov}\\,}}_{X \\sim p} (X)$ is the covariance matrix.\n\nAn upper bound of $\\psi (p)$ of the same form is relatively easy and it was shown to be achieved by half-spaces [12]. Proving the lower bound on $\\psi (p)$ up to some small factors in Conjecture 1 is the main goal of this paper. We say a log-concave density is isotropic if its mean ${\\mathbb {E}}_{X\\sim p} [X]$ equals to 0 and its covariance ${{\\,\\mathrm{Cov}}}_{X\\sim p}(X)$ equals to $\\mathbb {I}_d$.\nIn the case of isotropic log-concave densities, the KLS conjecture states that any isotropic log-concave density has its isoperimetric coefficient lower bounded by a universal constant.", |
| "content_sha256": "9b77b62dda98c75314bbe5ed3a44294e83d27d1396440074293e7e99fa60833d" |
| } |
| ], |
| "verification_status": "pending", |
| "verification_notes": null |
| }, |
| { |
| "evidence_id": "lee-vempala-nquarter-bound", |
| "reference_statement": "Lee and Vempala prove an O(n^{1/4}) upper bound on the Cheeger constant’s reciprocal for n-dimensional isotropic log-concave measures, improving the previously known O(n^{1/3} sqrt(log n)) bound.", |
| "scope_conditions": [ |
| "The measure is n-dimensional, isotropic, and log-concave", |
| "The statement is given in the paper’s convention for the Cheeger/KLS constant" |
| ], |
| "passages": [ |
| { |
| "source_id": "lee-vempala-2017", |
| "locator": "arXiv abstract, lines 16-18", |
| "text": "We show that the Cheeger constant for $n$-dimensional isotropic logconcave measures is $O(n^{1/4})$, improving on the previous best bound of $O(n^{1/3}\\sqrt{\\log n}).$ As corollaries we obtain the same improved bound on the thin-shell estimate, Poincaré constant and Lipschitz concentration constant and an alternative proof of this bound for the isotropic (slicing) constant; it also follows that the ball walk for sampling from an isotropic logconcave density in ${\\bf R}^{n}$ converges in $O^{*}(n^{2.5})$ steps from a warm start.", |
| "content_sha256": "3b5a7763416708f41ef60339b7cef6657418b961a34f9d7eecf13bb4c3cf8fc6" |
| } |
| ], |
| "verification_status": "pending", |
| "verification_notes": null |
| }, |
| { |
| "evidence_id": "chen-almost-constant-bound", |
| "reference_statement": "Chen proves a lower bound on the isoperimetric coefficient with dimension dependence d^{-o_d(1)}, and states that this improves the previous d^{-1/4} dependence for sufficiently large dimension.", |
| "scope_conditions": [ |
| "The result concerns log-concave densities and the KLS isoperimetric coefficient", |
| "The asymptotic improvement is stated for sufficiently large dimension", |
| "The d^{-o_d(1)} notation is the paper’s dimension-dependence formulation" |
| ], |
| "passages": [ |
| { |
| "source_id": "chen-2021", |
| "locator": "Section 2, Theorem 1 and its corollary, lines 118-131 of the Springer full text", |
| "text": "We prove the following lower bound on the isoperimetric coefficient of any log-concave density.\n### Theorem 1\nThere exists a universal constant c such that for any log-concave density p in $\\mathbb {R}^d$ and any integer $\\ell \\ge 1$, we have\n\n$$\\begin{aligned} \\psi (p) \\ge \\frac{1}{\\left[ c \\cdot \\ell \\left( \\log (d)+1 \\right) \\right] ^{\\ell /2} d^{16/\\ell } \\cdot \\sqrt{\\rho \\left( p \\right) }} \\end{aligned}$$\n\n(5)\n\nwhere $\\rho \\left( p \\right) $ is the spectral norm of the covariance matrix of p.\nAs a corollary, take $\\ell = \\left\\lceil \\left( \\frac{\\log (d)}{\\log \\log (d)} \\right) ^{1/2} \\right\\rceil $, then there exists a constant $c'$ such that\n\n$$\\begin{aligned} \\psi (p) \\ge \\frac{1}{d^{c' \\left( \\frac{\\log \\log (d)}{\\log {d}} \\right) ^{1/2}} \\cdot \\sqrt{\\rho \\left ( p \\right) }}. \\end{aligned}$$\n\nSince $\\lim _{d\\rightarrow \\infty } \\frac{\\log \\log (d)}{\\log (d)} = 0$, for $d$ large enough, the above lower bound is better than any lower bound of the form $\\frac{1}{d^{c''} \\sqrt{\\rho (p)} }$ ($c''$ is a positive constant) in terms of dimension $d$ dependency.", |
| "content_sha256": "57767dcd6f79a992627d712bd708226275b50323f15f7e118b8c874b198b7614" |
| } |
| ], |
| "verification_status": "pending", |
| "verification_notes": null |
| }, |
| { |
| "evidence_id": "klartag-sqrt-log-bound", |
| "reference_statement": "Klartag states that the Bourgain slicing conjecture and the KLS isoperimetric conjecture in R^n hold up to a factor of sqrt(log n).", |
| "scope_conditions": [ |
| "The statement is for the KLS isoperimetric conjecture in R^n", |
| "The abstract states an up-to-factor result rather than a dimension-free resolution", |
| "The source identifies an improved log-concave Lichnerowicz inequality as a proof ingredient" |
| ], |
| "passages": [ |
| { |
| "source_id": "klartag-2023", |
| "locator": "arXiv abstract, lines 0-4 and 22-24", |
| "text": "We prove that the Bourgain slicing conjecture and the Kannan-Lov\\'asz-Simonovits (KLS) isoperimetric conjecture in $\\varmathbb R^{n}$ hold true up to a factor of $\\sqrt{\\log n}$. A new ingredient used in the proof is an improved log-concave Lichnerowicz inequality.", |
| "content_sha256": "69799f77691fac47dc33f96c81610e4849f7495ecb0ed043b124fadab6b20f08" |
| } |
| ], |
| "verification_status": "pending", |
| "verification_notes": null |
| }, |
| { |
| "evidence_id": "thin-shell-not-kls-resolution", |
| "reference_statement": "Klartag and Lehec prove a universal thin-shell bound for isotropic log-concave random vectors, while their introduction explicitly characterizes thin-shell as historically easier than KLS; this result alone is therefore evidence for a related conjecture, not for the KLS conclusion.", |
| "scope_conditions": [ |
| "X is an isotropic, log-concave random vector in R^n", |
| "The universal bound concerns Var(|X|^2) and E(|X|-sqrt(n))^2", |
| "The paper’s historical comparison does not assert an implication from the proved thin-shell theorem to KLS" |
| ], |
| "passages": [ |
| { |
| "source_id": "klartag-lehec-2025-thin-shell-preprint", |
| "locator": "Theorem 1.1 and Corollary 1.2, HTML version lines 28-50; introductory discussion lines 60-64", |
| "text": "Let $X$ be an isotropic, log-concave random vector in $\\mathbb{R}^{n}$. Then,\n\n| $$\\mathop{\\mathrm{Var}}\\nolimits(|X|^{2})=\\mathbb{E}\\left(|X|^{2}-n\\right)^{2}\\leq Cn,$$ | | (1)\n\nwhere $C>0$ is a universal constant.\n\n...\n\nIf $X$ is log-concave and isotropic then\n\n| $$\\mathop{\\mathrm{Var}}\\nolimits(|X|)\\leq\\mathbb{E}(|X|-\\sqrt{n})^{2}\\leq C,$$ | | (3)\n\nwhere $C$ is a universal constant.\n\n...\n\nThus, for quite some time, the thin-shell conjecture was considered “harder” than the slicing problem but “easier” than the KLS conjecture.", |
| "content_sha256": "e4fdfc64f2ca3c31da4c9a71354338045bb3222ba2893ddfaf9cece84b20f62b" |
| } |
| ], |
| "verification_status": "pending", |
| "verification_notes": null |
| } |
| ] |
| } |
|
|
