BK: KLS - 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. The explicit BK Poincaré estimate implies the unchanged canonical KLS statement. The sole domain extension is to locally Lipschitz functions with infinite Dirichlet integral; the equivalent Cheeger assertion uses the established two-sided comparison.

Dependencies. The only theorem input is Theorem 11.2. For the equivalent Cheeger formulation use (0.3) Klartag, 2023Milman, 2009. Neither the existing status of Conjecture 0.1, another proof of KLS, nor a BKL or SZ v2 theorem is used. This composition does not certify its upstream theorem.

Proof 158.1

Fix $\mu$ and $f$ as stated. Isotropy makes the affine support all of $\mathbb R^n$ , and the law is absolutely continuous, so the almost everywhere gradient has a well-defined energy $E\in[0,\infty]$ . If $E<\infty$ , the explicit BK estimate gives $f\in L^2(\mu)$ and the claimed inequality with this single $C$ . For an $L^2$ function,

\int|f-c|^2d\mu=\operatorname{Var}_\mu f+ \left(c-\int f\,d\mu\right)^2,

(158.2)

so the extended convention agrees with ordinary variance. If $E=\infty$ , the inequality holds in the extended nonnegative reals. For completeness, a function outside $L^2$ cannot have $\int|f-c|^2d\mu<\infty$ at any finite $c$ , since $|f|^2\le2|f-c|^2+2c^2$ would contradict that fact. Thus no undefined expectation is being subtracted.

The Cheeger comparison gives $h_\mu^{-2}\le\pi C_P(\mu)\le\pi C$ , hence $h_\mu\ge(\pi C)^{-1/2}$ . Conversely a uniform lower bound $h_\mu\ge b>0$ implies $C_P(\mu)\le4h_\mu^{-2}\le4b^{-2}$ by the other side of the same comparison. These are precisely the two equivalent assertions of the canonical target.

Fences respected. This is a composition of the explicitly named BK input only. It establishes no sharper structural inequality or unrelated conditional premise. The explicitly named BK input is proved separately; this composition does not replace its proof or independent review.

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