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Math-73 · mathematics · Geometry

Kannan–Lovász–Simonovits Isoperimetric Conjecture

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Statement

Definitions and mathematical statement

For a log-concave probability density $e^{-V}$ on its affine support, $V$ is convex. Define the outer boundary measure by $\mu^+(B)=\liminf_{\varepsilon\downarrow0}(\mu(B_\varepsilon)-\mu(B))/\varepsilon$, and \[ h(\mu)=\inf_{0<\mu(B)<1}\frac{\mu^+(B)}{\min(\mu(B),1-\mu(B))}. \] The halfspace form asks for $h(\mu)\ge c\,h_{\rm half}(\mu)$, where the latter infimum is restricted to halfspaces and $c>0$ is universal. The dimension-free covariance formulation is $h(\mu)\ge c/\sqrt{\|\operatorname{Cov}(\mu)\|_{\rm op}}$. Work on the affine support for degenerate measures; point masses require the corresponding trivial convention.

Formulation and concept references: [S1].

Short English statement

For log-concave distributions, are halfspaces always within a universal factor of the worst possible isoperimetric bottleneck?

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Math-73-A2 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-73

The logarithmic Sobolev conclusion assumes a uniform subgaussian bound on every linear marginal. General log-concave laws need not have that bound at a covariance-comparable scale, so the unrestricted variance or KLS assertion here does not follow.

Kannan–Lovász–Simonovits Isoperimetric Conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 093; 2026-09-23), the asserted result is that a centered log-concave density with linear subgaussian parameter $a$ satisfies a logarithmic Sobolev inequality with constant bounded by an absolute multiple of $a^2$.

Mathematical statement of the claimed result

Manuscript 1.
Take any $n\ge1$, a centered log-concave probability measure $\mu$ on $\mathbb R^n$ with Lebesgue density, and $a>0$ such that $X\sim\mu$ obeys \[ \sup_{|\theta|=1}\mathbb E\exp(\langle X,\theta\rangle^2/a^2)\le2. \] The claimed absolute, dimension-independent constant $C$ gives, for all $f\in C_c^{\infty}(\mathbb R^n)$, \[ \operatorname{Ent}_{\mu}(f^2)\le Ca^2\int|\nabla f|^2\,d\mu, \qquad \operatorname{Ent}_{\mu}(g)=\int g\log g\,d\mu- (\int g\,d\mu)\log(\int g\,d\mu). \] Only linear-function subgaussianity is assumed, but a logarithmic Sobolev conclusion is claimed.

Outline of the argument

For manuscript 1 (family 093; 2026-09-23), gaussian regularization and entropy dissipation reduce the question to critical posterior laws. Efficient observation probes and nearly-first-eigenfunction hierarchies control the prediction curve, while curvature and operator-tail bounds rule out a dimension-dependent obstruction.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 093; 2026-09-23).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

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Math-73-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-73

Kannan–Lovász–Simonovits Isoperimetric Conjecture

Definitions and mathematical statement

For a log-concave probability density $e^{-V}$ on its affine support, $V$ is convex. Define the outer boundary measure by $\mu^+(B)=\liminf_{\varepsilon\downarrow0}(\mu(B_\varepsilon)-\mu(B))/\varepsilon$, and \[ h(\mu)=\inf_{0<\mu(B)<1}\frac{\mu^+(B)}{\min(\mu(B),1-\mu(B))}. \] The halfspace form asks for $h(\mu)\ge c\,h_{\rm half}(\mu)$, where the latter infimum is restricted to halfspaces and $c>0$ is universal. The dimension-free covariance formulation is $h(\mu)\ge c/\sqrt{\|\operatorname{Cov}(\mu)\|_{\rm op}}$. Work on the affine support for degenerate measures; point masses require the corresponding trivial convention.

Formulation and concept references: [S1].

Short English statement

For log-concave distributions, are halfspaces always within a universal factor of the worst possible isoperimetric bottleneck?

Research attempt

Scope, process, and checked context.
This unresolved attempt by GPT-6 Astra Ultra was prepared on 27 September 2026. The constant sought is independent of dimension and of the log-concave probability measure. All distances and covariance operators are Euclidean on the affine support; a point mass is separated as the trivial zero-dimensional case. The attack first proves the one-dimensional statement with explicit constants, then tries to pass from one-dimensional control to arbitrary cuts.

The primary localization source is Kannan--Lovasz--Simonovits [E1]; Bobkov [E2] explains related isoperimetric reductions. The recent manuscript [S2] states a logarithmic, dimension-dependent bound and a result for quadratic forms. The thin-shell result in [S4] controls a particular radial variance. Neither statement is substituted for the requested uniform bound on all measurable cuts. The calculations below use no claimed general solution of KLS.

A quantile calculation.
Let $f$ be a nondegenerate log-concave density on an interval of $\mathbb R$, with distribution function $F$ and median $m$. Put $c=f(m)>0$ and, for $0<u<1$, define \[ J(u)=f(F^{-1}(u)). \] The function $J$ is concave. Indeed, on the interior of the support $\log f$ is concave and locally Lipschitz, and the chain rule gives $J'(u)=(\log f)'(F^{-1}(u))$ almost everywhere. The right side is nonincreasing; the locally absolutely continuous function $J$ is therefore concave. This also covers nonsmooth log densities by one-sided derivatives. Endpoint limits are nonnegative, and concavity with $J(1/2)=c$ yields \begin{equation}\label{wsref-0-72-39-38-3a-70-72-6f-66-69-6c-65} J(u)\ge2c\min(u,1-u),\qquad \sup_{0<u<1}J(u)\le2c. \end{equation} For the upper bound, when $u<1/2$ apply concavity between $u$ and a point tending to $1$ to get $c\ge J(u)/(2(1-u))$; the other half is symmetric. These are not assertions that the density itself is symmetric or that its maximum occurs at the median.
Lemma 98.1: the exact one-dimensional Cheeger constant.
For this measure, \begin{equation}\label{wsref-0-72-39-38-3a-65-78-61-63-74} h(\mu)=2f(m). \end{equation} Here is a proof that includes arbitrary measurable cuts, not just half-lines. For any locally Lipschitz $g$, integration along intervals from $m$ and Tonelli's theorem give \begin{align*} \int |g(x)-g(m)|f(x)\,dx &\le \int_{t<m}|g'(t)|F(t)\,dt +\int_{t>m}|g'(t)|(1-F(t))\,dt\\ &\le \frac1{2c}\int |g'(t)|f(t)\,dt. \end{align*} The last inequality is exactly the lower bound in (r98:profile), on the appropriate side of the median.

For a measurable $B$, if its closure has strictly larger measure, $\mu(B_\varepsilon)-\mu(B)$ is bounded below by a positive constant and $\mu^+(B)=\infty$. Otherwise replace it by its closure without changing its measure or its open enlargements, and use \[ g_\varepsilon(x)= \max\{0,1-\operatorname{dist}(x,B)/\varepsilon\}. \] Its derivative vanishes almost everywhere off $B_\varepsilon\setminus B$, and has magnitude at most $1/\varepsilon$ there. Moreover $g_\varepsilon$ tends to $1_B$ in $L^1(\mu)$. For every $a\in[0,1]$, $\int|1_B-a|\,d\mu\ge\min(\mu(B),1-\mu(B))$; applying this with $a=g_\varepsilon(m)$ in the preceding functional inequality and taking a liminf proves \[ \mu^+(B)\ge2c\min(\mu(B),1-\mu(B)). \] The half-line $(-\infty,m]$ has mass $1/2$ and outer boundary measure $f(m)=c$, so it attains the bound and proves (r98:exact). The reasoning takes place on the support and also covers compactly supported densities.$\square$

Explicit comparison with variance.
Write $\sigma^2=\operatorname{Var}(X)$. Chebyshev gives mass at least $3/4$ to the interval of length $4\sigma$ centred at the mean. Consequently $\|f\|_\infty\ge3/(16\sigma)$. By (r98:profile), $\|f\|_\infty\le2c$, giving $h(\mu)\sigma\ge3/16$.

For the reverse bound, on $x\ge m$ the same profile estimate says $f(x)\ge2c(1-F(x))$. Solving the resulting differential inequality for the survival function, and doing the analogous calculation on the left, yields \[ \mathbb P(X\ge m+t)\le\tfrac12e^{-2ct},\qquad \mathbb P(X\le m-t)\le\tfrac12e^{-2ct}\quad(t\ge0). \] Thus \[ \sigma^2\le\mathbb E(X-m)^2 =\int_0^\infty2t\,\mathbb P(|X-m|>t)\,dt \le\frac1{2c^2}. \] Combining these inequalities gives the fully explicit, nonoptimal two-sided comparison \begin{equation}\label{wsref-0-72-39-38-3a-76-61-72-69-61-6e-63-65} \frac3{16}\le h(\mu)\sigma\le\sqrt2. \end{equation} No dimension parameter has appeared because this calculation is one-dimensional.

For the uniform measure on $[-a,a]$, the formula gives $h=1/a$ and $\sigma=a/\sqrt3$. For the density $f(x)=e^{-|x|/b}/(2b)$ it gives $h=1/b$ and $\sigma=\sqrt2b$, attaining the upper constant above. For the one-sided exponential $f(x)=e^{-x}$ on $[0,\infty)$, the median is $\log2$ and $h=1$, even though the density maximum is at zero. These examples check symmetry, support boundaries and normalization.

What this proves about halfspaces in every dimension.
Let $\Sigma$ be positive definite on the affine support and put $s=\sqrt{\|\Sigma\|_{\rm op}}$. The marginal theorem of Prekopa [E3] for log-concave densities makes the law of $\langle\theta,X\rangle$ log-concave for every unit vector $\theta$. Its variance is $\sigma_\theta^2=\theta^{\mathsf T}\Sigma\theta$. For a halfspace perpendicular to $\theta$, Euclidean enlargement translates its threshold by exactly $\varepsilon$, so its boundary ratio is the corresponding one-dimensional ratio. By (r98:exact) its infimum over thresholds is attained at the median. Therefore (r98:variance) gives \begin{equation}\label{wsref-0-72-39-38-3a-68-61-6c-66} \frac3{16s}\le h_{\rm half}(\mu)\le\frac{\sqrt2}{s}. \end{equation} The lower bound uses $\sigma_\theta\le s$ in every direction; the upper bound chooses an eigenvector for the largest eigenvalue.

This verifies the constant-factor relationship between the two catalogue formulations. If $h\ge c_0h_{\rm half}$, then $h\ge3c_0/(16s)$. Conversely, $h\ge c_1/s$ and the upper bound in (r98:half) give $h\ge(c_1/\sqrt2)h_{\rm half}$. What has not been proved is either premise for arbitrary sets. The elementary inequality from taking an infimum over more sets goes in the other direction, $h\le h_{\rm half}$.

A precise obstruction to the naive localization step.
It is tempting to apply the one-dimensional lemma to every needle arising in localization and declare its variance bounded by isotropy of the original measure. That declaration is false even for a cube. Let $\mu$ be uniform on $[-\sqrt3,\sqrt3]^n$. Its covariance is $I$. Along the diagonal unit vector $\theta=(1,\ldots,1)/\sqrt n$, the central chord consists of $t\theta$ with $|t|\le\sqrt{3n}$. The uniform probability measure on this chord has \[ \operatorname{Var}(t)=n,\qquad h=\frac1{\sqrt{3n}}. \] It is a legitimate one-dimensional log-concave measure on a segment inside the same body. By contrast, the projection of the original cube measure onto $\theta$ has variance one. A restriction to a chord and a marginal are different probability measures.

This does not say that a particular localization construction must select that chord, or that the cube violates KLS. It disproves the proposed justification that every possible needle inherits the ambient covariance bound. A localization proof must preserve appropriate moment constraints or otherwise control the bad needles; the one-dimensional theorem alone supplies neither.

There is also a basic scaling check. Under $x\mapsto ax+b$ with $a>0$, neighbourhood radii scale by $a$, so $h$ scales by $a^{-1}$ and $s$ by $a$. For a general invertible linear map $A$, if its smallest and largest singular values are $s_{\min},s_{\max}$, the inclusions between transformed neighbourhoods give \[ \frac{h(\mu)}{s_{\max}} \le h(A_*\mu)\le\frac{h(\mu)}{s_{\min}}. \] Thus whitening is useful but does not leave the Euclidean boundary constant unchanged. All appearances of isotropy must account for that metric change.

Verification and first gap.
The exact one-dimensional formula was proved through a functional inequality and distance cutoffs, including nonsmooth cuts. The variance bounds use only concavity and elementary tail integration; the examples and diagonal-chord calculation are exact. The marginal step imports preservation of log-concavity under linear projection, but it makes no claim about conditional needle variances. These are classical ingredients and transparent deductions, with no novelty claim or numerical evidence of a general solution.

The first missing inequality is a dimension-independent comparison between arbitrary-cut boundary ratios and the halfspace ratios in (r98:half). Next targets are a localization procedure with controlled needle variances, an all-functions functional inequality strong enough to test distance cutoffs, and a search for isotropic log-concave examples whose nonlinear cuts outperform halfspaces by an unbounded factor. Quadratic or radial tests alone do not cover those distance functions, and the diagonal chord is not a counterexample to the original measure's inequality.

Final status.
UNRESOLVED. The one-dimensional formula and variance bounds, the halfspace/covariance comparison, and the failure of a naive needle argument are verified. General isoperimetric control in arbitrary dimension remains unproved.

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