Song–Zhang v2: KLS - The KLS theorem and its methods

Part of the second version of Song–Zhang, Chapter Song–Zhang, second version: repeated refinement with summable losses; the reading order is on the full proofs page.

Overview. This dossier composes Theorem 9.3 with the unchanged canonical target Conjecture 0.1. The only domain adjustment is to interpret the Poincaré inequality for locally Lipschitz functions whose Dirichlet integral is infinite. The Cheeger formulation then follows from the established two-sided comparison.

Dependencies. The only theorem input is Theorem 9.3. For the equivalent Cheeger formulation we use (0.3), with the normalization and references recorded there Klartag, 2023Milman, 2009. There is no BKL premise, no consequence of BKL, and no invocation of the already recorded status of Conjecture 0.1. The upstream Song–Zhang reconstruction supplies its own universal Poincaré theorem. This composition does not certify that upstream theorem or substitute for its independent review.

Proof 147.1

Let $C$ be the universal constant of Theorem 9.3, and fix $n$ , $\mu$ and $f$ as in the statement. The isotropic assumption implies that the affine hull of the support is all of $\mathbb R^n$ ; a full-dimensional log-concave measure has a Lebesgue density. Hence the a.e. gradient of the locally Lipschitz function is defined $\mu$ -a.e. and its squared norm has a well-defined integral in $[0,\infty]$ .

If $E=\int|\nabla f|^2d\mu$ is finite, Theorem 9.3 applies with exactly this function and measure. It gives both $f\in L^2(\mu)$ and $\operatorname{Var}_\mu(f)\le CE$ . For $f\in L^2$ , expansion of the square gives

\int|f-c|^2d\mu =\int|f-\textstyle\int f\,d\mu|^2d\mu +\bigl(c-\textstyle\int f\,d\mu\bigr)^2,

(147.2)

so the displayed extended convention equals ordinary variance. If $E=\infty$ , the right-hand side of the claimed inequality is $\infty$ , and the inequality holds in the extended nonnegative reals. This includes nonintegrable functions without attempting to subtract an undefined mean. Indeed if $\int|f-c|^2d\mu$ were finite for any finite $c$ , the inequality $|f|^2\le2|f-c|^2+2c^2$ would imply $f\in L^2(\mu)$ . Thus the extended variance is $\infty$ for every function outside $L^2$ , as required. The same single constant $C$ works in every dimension and for every allowed measure and function.

It follows that $C_P(\mu)\le C$ for all isotropic log-concave $\mu$ . In the manuscript convention $\Psi_{\mathrm{KLS},\mu}=h_\mu^{-1}$ , (0.3) gives

h_\mu^{-2}\le\pi C_P(\mu)\le\pi C, \qquad h_\mu\ge(\pi C)^{-1/2}.

(147.3)

Conversely, if $h_\mu\ge c>0$ uniformly, the other side of that same comparison gives $C_P(\mu)\le4h_\mu^{-2}\le4c^{-2}$ . This is precisely the equivalence asserted in the canonical target.

Fences respected. This composition claims only a universal constant. It establishes no sharper structural inequality and does not discharge any unrelated conditional premise. Its proof uses the Song–Zhang theorem and the earlier Cheeger comparison, rather than the other proof record attached to the same target.

References¶

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