BK: the Poincaré and Cheeger constants - The KLS theorem and its methods

Part of the Balasubramanian–Kasiviswanathan proof, Chapter Balasubramanian–Kasiviswanathan: compatible integration; the reading order is on the full proofs page.

Overview. Fix a regular measure and let the number of observed Appell degrees tend to infinity. The integration-power bound loses its curvature dependence. Approximation then preserves the resulting scalar Poincaré constant. A contraction from isotropic coordinates handles every covariance at most the identity, including proper affine supports. The existing Cheeger comparison converts the spectral constant to the manuscript’s inverse Cheeger normalization. This reconstructs Corollary 8.2, Section 9 and Appendix G of Balasubramanian & Kasiviswanathan, 2026, pinned at commit 4837c33649ba2271f43c9684e9350ecbdd725f95.

Dependencies. The substantive inputs are Theorem 11.1, Corollary 11.1, and Lemma 11.1, in the conventions Definition 11.1 and Definition 11.2. For the Cheeger corollary only, use the classical comparison (0.3) Klartag, 2023Milman, 2009. The argument assumes these BK inputs as stated; it neither reconstructs nor certifies them here. It does not use the existing status of Conjecture 0.1, a BKL result, or any SZ v2 result.

Theorem 157.1 (Explicit universal bounds)

Assume the BK inputs listed above. Put $R=10^8$ and $K_P=1+2R^2=1+2\cdot10^{16}$ . Every log-concave probability law $\mu$ of covariance at most $I$ , in every dimension, satisfies

\operatorname{Var}_\mu f\le K_P\int|\nabla f|^2\,d\mu

(157.1)

for every real locally Lipschitz function on its affine support with finite Dirichlet energy. Every such function belongs to $L^2(\mu)$ . Gradients are intrinsic to that support, and a point mass has Poincaré constant zero. For every isotropic log-concave law of positive dimension,

h(\mu)^{-1}\le\sqrt{\pi(1+2\cdot10^{16})}.

(157.2)

These are Theorem 11.2 and Corollary 11.3.

Proof 157.1

Fix a centered regular law $\nu$ with covariance at most $I$ and fix one positive lower curvature bound $a$ for it. The uniform coefficient theorem gives $c_j(\nu)\le R^j/(j+1)^4$ for every $j\ge1$ . For each integer $D\ge2$ , the integration-power corollary, with $\rho=R$ , implies

C_P(\nu)\le1+2R^2\max\{1,a^{-1/(D+1)}\}.

(157.3)

Here the measure and $a$ remain fixed. Letting $D\to\infty$ yields $C_P(\nu)\le1+2R^2=K_P$ . Thus the scalar constant is independent of both Hessian bounds and of dimension.

Now let $\mu$ be isotropic and log-concave on $\mathbb R^n$ . The regular approximation lemma supplies regular isotropic $\mu_j$ converging weakly to $\mu$ . Each satisfies the just-proved Poincaré bound $K_P$ . For $u\in C_c^\infty$ , weak convergence passes its variance and Dirichlet energy to the limit, since $u,u^2,|\nabla u|^2$ are bounded and continuous. The finite-energy extension in the same approximation lemma then gives the inequality and square integrability for every locally Lipschitz finite-energy test under $\mu$ . No uniform curvature bound for $\mu_j$ is required.

For an arbitrary covariance $\Sigma\preceq I$ , translate to center the law and work on its linear support $S$ . If $\dim S=0$ there is nothing to prove: every real function is constant almost surely. Otherwise $\Sigma$ is positive definite on $S$ . Indeed a zero variance direction would force the centered law onto a smaller linear subspace. Set $Y=\Sigma^{-1/2}X$ on $S$ , which is isotropic and log-concave. For a locally Lipschitz function $f$ on $S$ of finite energy, the function $g(y)=f(\Sigma^{1/2}y)$ is locally Lipschitz and

\int|\nabla g|^2d\mathcal L(Y) =\mathbb E\langle\nabla f(X),\Sigma\nabla f(X)\rangle \le\mathbb E|\nabla f(X)|^2<\infty.

(157.4)

The chain rule holds Lebesgue almost everywhere on $S$ , hence almost surely under this full-dimensional log-concave law on $S$ . The isotropic result implies $g(Y)\in L^2$ and

\operatorname{Var}f(X)=\operatorname{Var}g(Y) \le K_P\mathbb E\langle\nabla f(X),\Sigma\nabla f(X)\rangle \le K_P\mathbb E|\nabla f(X)|^2.

(157.5)

Translation back to the original affine support changes neither variance nor energy.

Finally, in the normalization $\Psi_\mu=h(\mu)^{-1}$ , the existing comparison gives $h(\mu)^{-2}\le\pi C_P(\mu)\le\pi K_P$ . Taking the nonnegative square root proves the Cheeger assertion.

Fences respected. The degree limit is taken at one fixed regular law, then the scalar inequality is passed to nonsmooth laws. The covariance contraction is intrinsic to the affine support. The Cheeger constant is exactly the convention of (0.5), not its reciprocal. No sharper thin-shell, moment-map, conditional-fiber, or occupation premise is claimed.

References¶

Balasubramanian, K., & Kasiviswanathan, S. (2026). A Dimension-Free Bound on the Poincaré Constant of Isotropic Log-Concave Measures. https://arxiv.org/abs/2610.07728v1
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