The weighted near-Cheeger implication - The KLS theorem and its methods

Part of the fixed-cut archive, Chapter The fixed cut: approach and lessons; the reading order is on the full proofs page.

Overview. The weighted package supplies an integrated Stein-trace estimate and bounds its error by a multiple of time for cuts whose initial excess is at most one. The certified source conversion then gives an absorptive Carleson premise on the package’s fixed window. Tight-window consumption forces a uniform positive perimeter for balanced near-minimizers, which implies KLS by the certified half-mass identity. The package itself is not proved here; its recorded refutation is not used to make the implication vacuous.

Proof 102.1

Retain the fixed window parameter $\eta$ and constants $\beta,\gamma$ of the package. Write

\alpha=2\beta+64\eta^2<1,\qquad T_1=\min\{T_0,1\}>0.

(102.1)

Since $\beta\ge0$ , its absorption margin implies $\eta<1/8<1/6$ . Thus the package’s window is already admissible for Corollary 30.1; it need not and must not be changed after the assumptions have been supplied.

Fix an isotropic log-concave $\mu$ and a finite-perimeter cut $E$ with $p_0=\mu(E)=1/2$ and $e_0(E)\le1$ . Let $\tau_\eta$ be the fixed tight exit time. Use the notation $r,S,D,s$ of the manuscript and put

W_t=(1+\|A_t\|_{\mathrm{op}})^{5/2},\qquad \mathcal S_t=\mathcal S_{\mu_t}(E).

(102.2)

On $\{t<\tau_\eta\}$ , Lemma 32.1 gives $S_t\le2\mathcal S_t/s_t+64\eta^2D_t$ . Integrate this nonnegative pointwise inequality and apply clause (ii-w) of the package. For every $0<T\le T_1$ this yields

\begin{aligned} \mathbb E\int_0^{T\wedge\tau_\eta}S_t\,dt &\le2C_0T+2C_1\mathbb E\int_0^{T\wedge\tau_\eta}r_t\,dt +\alpha\mathbb E\int_0^{T\wedge\tau_\eta}D_t\,dt\\ &\quad+2C_2\mathbb E\int_0^{T\wedge\tau_\eta}e_t(E)W_t\,dt. \end{aligned}

(102.3)

No damping term has been subtracted in this step. Clause (i-w), $e_0\le1$ , $\gamma>0$ , and $T\le1$ give

2C_2\mathbb E\int_0^{T\wedge\tau_\eta}e_t(E)W_t\,dt \le2C_2^2(Te_0+T^{1+\gamma})\le4C_2^2T.

(102.4)

Consequently the source satisfies exactly the prefix premise of Corollary 30.1, with universal data

C'_0=2C_0+4C_2^2,\qquad C'_1=2C_1,\qquad \alpha<1,\qquad T_1>0,

(102.5)

and the unchanged universal $\eta$ . The initial nested-window condition is automatic because $p_0=1/2$ . The certified corollary supplies a constant $c_*>0$ , depending only on these universal data, such that

\mu^+(E)\ge c_*

(102.6)

for every such pair $(\mu,E)$ .

To pass from these cuts to the Cheeger constant, fix $\mu$ and let $I=I_\mu(1/2)$ . If $I=+\infty$ , the required lower bound follows directly from Lemma 33.1. Otherwise, by the definition of the infimum defining the profile, choose measurable half-mass sets $E_k$ of finite perimeter with

I\le\mu^+(E_k)\le I+1/k.

(102.7)

These sets have $0\le e_0(E_k)\le1/k\le1$ , so the preceding argument applies to each of them. Letting $k$ tend to infinity gives $I\ge c_*$ . Finally Lemma 33.1 gives $h_\mu=2I\ge2c_*$ . The constant is universal, which is the asserted KLS conclusion.

Dependencies and fences. The proof uses Lemma 32.1, Corollary 30.1, and Lemma 33.1, with Assumption 28.3 as its antecedent. The target has no bounded_by edge. The tight-window result is used as a certified implication, including its analytical consumption step, rather than reproved here. The two-tail and spectator obstructions are respected: no slice-wise estimate beyond the certified conversion is asserted, no global covariance weight is claimed to be valid, and the refuted package is not discharged. The implication provides no unconditional progress on KLS.