Letwin: the bound $\CP\lesssim\sqrt{\log n}$ - The KLS theorem and its methods

Part of the results of the literature written out here, Chapter Family 4: moment maps, Monge–Ampère, and Stein kernels; the reading order is on the full proofs page.

Overview. The third-moment bound gives a fixed-time covariance window. A Gaussian observation transfers the variance of each Lipschitz test to its posterior. The improved Lichnerowicz inequality bounds the posterior Poincaré constant. Choosing the observation time below both the covariance window and the initial spectral gap closes the estimate without assuming the desired bound on that gap. A published regular approximation then removes the smoothness restriction.

Sources and exact inputs. The claim is Theorem 1.1 of Letwin, 2026, pinned to arXiv:2607.24164v1. Its quadratic input is supplied here by Theorem 25.1, whose proof is in the separate dossier; the present dossier reconstructs the remaining passage to the general Poincaré constant. The published sources used in this passage are:

Klartag & Lehec, 2022, Corollary 5.4, checked in arXiv:2203.15551v2, Section 5: universal constants $a,b>0$ give $\mathbb E\|A_t\|_{\mathrm{op}}\le b$ for $0<t\le a/(\kappa_n^2\log n)$ . Only its moment exponent $p=1$ is used.
Klartag, 2023, checked in the published-version PDF arXiv:2303.14938v2, DOI Klartag (2023): Theorem 1.3 is Theorem 3.1; Lemma 2.1 supplies regular approximation. The variance-transfer mechanism of Lemma 3.1 and Corollary 3.2 is written out below, including its restriction on time.
The published Lipschitz-variance comparison of Milman, 2009, in the explicit normalization (3.10) of the preceding PDF, gives a universal $c_M>0$ such that $c_M\CP(\nu)\le\sup_{\operatorname{Lip}(f)\le1}\operatorname{Var}_\nu f$ for log-concave $\nu$ . This comparison is an imported theorem, not a consequence of the elementary Poincaré inequality in the other direction.

The constants $a,b,c_M$ do not depend on dimension, the initial law, its regularization parameters, or the test function. They need not be optimal. The published theorem Theorem 26.2 is not used as an input.

Proof 108.1

The third-moment parameter. Use exactly the parameter of Proposition 26.1:

\kappa_n=\sup_{\nu\text{ isotropic log-concave on }\mathbb R^n} \sup_{|u|=1} \left\|\mathbb E_\nu[\langle X,u\rangle X\otimes X]\right\|_{\mathrm{HS}}.

(108.2)

The source’s Schatten two-norm is the Hilbert--Schmidt norm; fixing the direction to the first coordinate gives the same supremum by orthogonal invariance. In particular the expectation is taken before the matrix norm. The antecedent of Proposition 26.1 is discharged by Theorem 25.1, giving $\kappa_n\le2\sqrt2$ . Also $\kappa_n\ge2$ : take independent $X_i=E_i-1$ , where each $E_i$ has density $e^{-s}\mathbf1_{s>0}$ . These variables have mean zero, variance one and third moment two. Independence makes $\mathbb E[X_1X\otimes X]=2e_1\otimes e_1$ .

The regular class and posterior. Initially let $\mu$ be isotropic and regular in the sense of Klartag’s Section 2: its density $e^{-U}$ is smooth and positive on $\mathbb R^n$ , with $\varepsilon I\preceq\nabla^2U\preceq\varepsilon^{-1}I$ for some $\varepsilon>0$ , and all derivatives of $U$ have at most polynomial growth. Write $P=\CP(\mu)$ . Then $1\le P<\infty$ : coordinate tests give the lower bound and strong convexity gives $P\le1/\varepsilon$ .

Fix $t>0$ and let $Y=tX+\sqrt t G$ , with $X\sim\mu$ and an independent standard Gaussian $G$ in $\mathbb R^n$ . Bayes’ formula gives the conditional law of $X$ given $Y=y$ as

\mu_{t,y}(dx)=Z(t,y)^{-1} e^{\langle y,x\rangle-t|x|^2/2}\mu(dx).

(108.3)

The normalizer is positive and finite. Denote its mean by $m(t,y)$ and its covariance by $A(t,y)$ . The localization covariance $A_t$ has the same law as $A(t,Y)$ ; this is the observation representation (3.1)--(3.5) of Klartag’s Section 3. No assertion about a supremum over times is needed. The posterior Hessian is $\nabla^2U+tI\succeq tI$ , so the ordinary and improved Lichnerowicz inequalities give, for every $y$ ,

A(t,y)\preceq t^{-1}I, \qquad \CP(\mu_{t,y})\le\sqrt{\|A(t,y)\|_{\mathrm{op}}/t}.

(108.4)

The first inequality follows by applying $\CP(\mu_{t,y})\le1/t$ to linear tests; the second is Theorem 3.1. These applications are to full-space smooth positive densities and introduce no boundary condition.

Variance transfer, with its time factor. For a globally 1-Lipschitz $f$ , put $F(y)=\int f\,d\mu_{t,y}$ . All relevant moments are finite. Differentiation under the integral is justified on compact sets of $y$ by the Gaussian factor in the likelihood and the at-most-linear growth of $f$ . It yields

\nabla F(y)=\operatorname{Cov}_{\mu_{t,y}}(X,f), \qquad |\nabla F(y)|^2\le \|A(t,y)\|_{\mathrm{op}}\operatorname{Var}_{\mu_{t,y}}f.

(108.5)

The latter inequality is Cauchy--Schwarz after pairing the vector with each unit direction. Tensorization and the chain rule give $\CP(Y)\le t^2P+t$ . Explicitly, conditional variance first in $G$ and then in $X$ bounds the variance of $g(tX+\sqrt tG)$ by $(t+t^2P)\mathbb E|\nabla g(Y)|^2$ ; Jensen bounds the derivative of the conditional mean in the second term. Smooth approximation extends this inequality to functions with finite Dirichlet energy. It applies to $F$ : conditional expectation is an $L^2$ contraction, and the preceding gradient estimate and covariance cap give $\mathbb E|\nabla F(Y)|^2\le t^{-1}\mathbb E \operatorname{Var}_{\mu_{t,Y}}f<\infty$ . Consequently

\operatorname{Var}(F(Y)) \le(t^2P+t)\mathbb E|\nabla F(Y)|^2 \le(1+tP)\mathbb E\operatorname{Var}_{\mu_{t,Y}}f.

(108.6)

The conditional-variance identity now gives

\operatorname{Var}_\mu f =\mathbb E\operatorname{Var}_{\mu_{t,Y}}f+\operatorname{Var}(F(Y)) \le(2+tP)\mathbb E\operatorname{Var}_{\mu_{t,Y}}f.

(108.7)

As $|\nabla f|\le1$ almost everywhere, the posterior Poincaré inequality and then the improved bound imply

\operatorname{Var}_\mu f \le\frac{2+tP}{\sqrt t}\, \mathbb E\sqrt{\|A_t\|_{\mathrm{op}}}.

(108.8)

Taking the supremum over globally 1-Lipschitz tests and using the published comparison with constant $c_M$ gives the precise bridge

c_MP\le\frac{2+tP}{\sqrt t} \mathbb E\sqrt{\|A_t\|_{\mathrm{op}}}.

(108.9)

In particular, a uniform numerical bound on $2+tP$ is required. This is the time restriction in the published Corollary 3.2, rather than an unconditional estimate at every point of the covariance window.

Closing the time restriction. Set

T=\frac{a}{\kappa_n^2\log n}, \qquad t=\min\{T,P^{-1}\}, \qquad K=\frac{3\sqrt b}{c_M}.

(108.10)

These times are strictly positive because $n\ge2$ and $2\le\kappa_n\le2\sqrt2$ . The published covariance estimate applies at $t\le T$ . Jensen’s inequality gives $\mathbb E\sqrt{\|A_t\|_{\mathrm{op}}}\le\sqrt b$ . Since $tP\le1$ , the bridge becomes $P\le K/\sqrt t$ . If $T\le P^{-1}$ this yields $P\le K/\sqrt T$ . If $P^{-1}\le T$ it yields $P\le K\sqrt P$ , hence $P\le K^2$ . Equality of the two times causes no problem, so in all cases

P\le\max\left\{K^2,\frac K{\sqrt a}\kappa_n\sqrt{\log n}\right\} \le D\kappa_n\sqrt{\log n}, \qquad D=\max\left\{\frac{K^2}{2\sqrt{\log2}},\frac K{\sqrt a}\right\}.

(108.11)

The last step uses $\kappa_n\sqrt{\log n}\ge2\sqrt{\log2}$ . Thus $D$ is universal. This choice of time uses no a priori quantitative bound for $P$ and no near-maximizer of $C_{\mathrm P,n}$ or $\PsiKLS_n$ .

Removal of regularity. Let now $\mu$ be any isotropic log-concave probability on $\mathbb R^n$ . Its affine support is all of $\mathbb R^n$ because its covariance is $I$ , so it is absolutely continuous. Its Poincaré constant is finite in each fixed dimension by the qualitative log-concave Poincaré theorem Bobkov, 1999; no estimate of that constant is used. Apply the published Lemma 2.1 of Klartag, 2023: for every $0<\delta<1$ there is a regular log-concave law $\nu_\delta$ with

\CP(\nu_\delta)\ge\CP(\mu)-\delta, \qquad \|\operatorname{Cov}(\nu_\delta)-I\|_{\mathrm{op}}<\delta.

(108.12)

The lemma’s additional assertion about preserving a positive curvature lower bound is not needed here. Write $m_\delta$ and $S_\delta$ for the mean and covariance of $\nu_\delta$ . Since $S_\delta\succ0$ , the affine image under $x\mapsto S_\delta^{-1/2}(x-m_\delta)$ is isotropic and still regular: the Hessian transforms by congruence, its two positive bounds remain finite, and polynomial growth of derivatives is preserved by a fixed invertible affine map. The regular-case estimate applies to this image. Pulling its Poincaré inequality back by the chain rule gives

\CP(\mu)-\delta\le\CP(\nu_\delta) \le\|S_\delta\|_{\mathrm{op}}D\kappa_n\sqrt{\log n} \le(1+\delta)D\kappa_n\sqrt{\log n}.

(108.13)

Here the same $\kappa_n$ bounds every isotropic approximant; no continuity of $\kappa_n$ , eigenfunctions, or posterior paths is required. Letting $\delta\downarrow0$ proves the bound for $\mu$ , including uniform laws on convex bodies and nonsmooth convex potentials. The Poincaré constant throughout is the constant for all locally Lipschitz tests, not just the globally Lipschitz tests used to estimate it.

Finally $\kappa_n\le2\sqrt2$ implies $C_{\mathrm P,n}\le2\sqrt2D\sqrt{\log n}$ . The reverse-Cheeger direction of (0.3), namely $\PsiKLS_\mu^2\le\pi\CP(\mu)$ , then gives the inverse-Cheeger bound. The single constant $C=\max\{2\sqrt2D,\sqrt{2\sqrt2\pi D}\}$ works in both inequalities. All logarithms are natural.

Hypotheses and fences. Isotropy identifies the initial covariance and the dimensionwise parameter $\kappa_n$ . Log-concavity is used in the covariance theorem, the posterior curvature bound, the Lipschitz-variance comparison and regular approximation. The restriction $n\ge2$ keeps the window positive and permits absorption of the constant branch. Regularity is an intermediate hypothesis removed in the proof. The node has no registered bounded_by edge. Against the program’s general fences: this dimension-dependent bound does not settle Conjecture 0.1; the proof never upgrades a fixed-time moment to a supremum-over-time event; it makes no trace-to-operator substitution; and it uses no cut-relative source, moving competitor, gate-zero, or CMH premise. The only preprint input is the separately reviewed Theorem 25.1, through Proposition 26.1. No additional conditional premise or numerical experiment is used.

References¶

Letwin, B. (2026). The KLS Constant is O(\log1/4 n).
Klartag, B., & Lehec, J. (2022). Bourgain’s Slicing Problem and KLS Isoperimetry up to Polylog. Geometric and Functional Analysis, 32(5), 1134–1159. 10.1007/s00039-022-00612-9
Klartag, B. (2023). Logarithmic Bounds for Isoperimetry and Slices of Convex Sets. Ars Inveniendi Analytica, 2023(4), 1–17. 10.15781/jsjy-0b06
Milman, E. (2009). On the Role of Convexity in Isoperimetry, Spectral Gap and Concentration. Inventiones Mathematicae, 177(1), 1–43. 10.1007/s00222-009-0175-9
Bobkov, S. G. (1999). Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures. Annals of Probability, 27(4), 1903–1921. 10.1214/aop/1022874820