The fixed cut: the bootstrap

Part of the fixed-cut archive (Chapter The fixed cut: approach and lessons); this chapter holds the bootstrap comparison theorem, which reduces excess propagation for near-worst measures to the single quantity $h_\mu\,\Xi_T(\mu)$ of (28.8), and the evaluation of $\Xi_T$ against the known covariance estimates.

Two elementary lemmas¶

Proof 33.1

Milman’s Theorem 1.8 and Corollaries 6.5 and 6.12 Milman, 2009 give symmetry and concavity of the profile on $(0,1)$ , including for nonsmooth log-concave laws in dimension one. The assigned endpoint value $I(0)=0$ need not equal $I(0^+)$ : the uniform law on an interval of length $L$ has $I(p)=1/L$ for $0<p<1$ .

For $0<\delta<p<1/2$ , interior concavity and nonnegativity give

I(p)\ge\frac{1/2-p}{1/2-\delta}I(\delta) +\frac{p-\delta}{1/2-\delta}I(1/2) \ge\frac{p-\delta}{1/2-\delta}I(1/2).

(33.2)

Letting $\delta\downarrow0$ yields $I(p)\ge2pI(1/2)$ ; symmetry handles $p>1/2$ . Equality of the infimum follows by taking $p=1/2$ , so $h_\nu=\inf_{0<p<1}I(p)/\min(p,1-p)=2I(1/2)$ . For proper affine support, apply this argument in the affine hull; a Dirac law has no nontrivial-mass competitors. At mass $1/2$ , the Cheeger line equals the profile, giving the excess identity. No endpoint continuity or profile minimizer is needed.

Proof 33.2

Let $\tilde\nu=A_\nu^{-1/2}{}_\#(\nu-a_\nu)$ , isotropic log-concave, so $h_{\tilde\nu}\ge\hstar_n$ , and $\nu=T_\#\tilde\nu+a_\nu$ with $T=A_\nu^{1/2}$ , $\norm T_\op=\lmax^{1/2}$ . For Borel $S$ and $\eps>0$ , $T^{-1}(S_\eps)\supseteq(T^{-1}S)_{\eps/\norm T_\op}$ , whence $\nu^+(S)\ge\tilde\nu^+(T^{-1}S)/\norm T_\op\ge(\hstar_n/\norm T_\op)\min(\nu(S),1-\nu(S))$ . For monotonicity: if $\nu$ is isotropic log-concave on $\R^m$ , $m\le n$ , then $\nu\otimes\gamma_{n-m}$ is isotropic log-concave on $\R^n$ and cylinder sets give $h_{\nu\otimes\gamma_{n-m}}\le h_\nu$ , so $\hstar_n\le\hstar_m$ .

The main inequality¶

Theorem 33.1 (Bootstrap comparison — Proved · agent review (gpt-6-astra, 2026-10-04) · thm:bootstrap · Correction)

Let $n\ge2$ , $\eps\in(0,1]$ , and let $\mu$ be isotropic log-concave on $\R^n$ with $h_\mu\le(1+\eps)\hstar_n$ (a near-worst measure of its dimension). Let $E$ be any measurable set with $\mu(E)=\tfrac12$ , $e_0=e_0(E)=\bar e_0(E)$ , and let $\eta\in(0,\tfrac12)$ . Then, with $X_t=(\lmax(A_t)-1)_+$ as in (28.8), for every $t>0$ :

\E\bigl[\bar e_t(E)\,\one_{\{t<\tau_\eta\}}\bigr] \ \le\ e_0+h_\mu\Bigl(\frac\eps2+\eta+\frac12\,\Prob(\tau_\eta\le t)+\frac14\,\E X_t\Bigr),

(33.4)

\Prob(\tau_\eta\le t)\ \le\ \frac1{4\eta^2}\Bigl(t+\int_0^t\E X_s\dd s\Bigr).

(33.5)

Consequently, with $\Xi_T=\int_0^T\E X_t\dd t$ ,

\int_0^T\E\bigl[\bar e_t\,\one_{\{t<\tau_\eta\}}\bigr]\dd t \ \le\ T e_0+h_\mu\Bigl[\Bigl(\frac\eps2+\eta\Bigr)T +\frac{T^2}{16\eta^2}+\frac{T\,\Xi_T}{8\eta^2}+\frac{\Xi_T}{4}\Bigr],

(33.6)

and, for $0<T<1/8$ , the admissible choice $\eta=T^{1/3}<1/2$ together with $\eps\le T^{1/3}$ gives the clean form

\int_0^T\E\bigl[\bar e_t\,\one_{\{t<\tau_\eta\}}\bigr]\dd t \ \le\ Te_0+C\,h_\mu\bigl(T^{4/3}+\Xi_T\bigr).

(33.7)

Idea of the proof. The perimeter supermartingale bounds the stopped perimeter by $\mu^+(E)$ ; whitening plus the elementary bound $\lambda^{-1/2}\ge1-\tfrac12(\lambda-1)_+$ converts that into a Cheeger statement on the balanced event, with the loss charged to $X_t$ ; near-worstness compares $h_{\mu_t}$ with $h_\mu$ ; and the exit probability of the tight window is bounded through the quadratic variation of $p_t$ , again by $X_t$ . The five steps are written out in Section Proof of the bootstrap comparison.

Theorem 33.2 (Stopped covariance-interface refinement — Proved · agent review (2026-08-27) · thm:bootstrap-stopped-interface · Correction)

Under the hypotheses of Theorem 33.1, define

\widehat\Xi_{T,\eta}(\mu,E) :=\int_0^T\E\bigl[X_s\one_{\{s<\tau_\eta\}}\bigr]\dd s.

(33.8)

Then

\begin{aligned} \int_0^T\E\bigl[\bar e_t(E)\one_{\{t<\tau_\eta\}}\bigr]\dd t \le Te_0+h_\mu\biggl[ &\Bigl(\frac\eps2+\eta\Bigr)T+\frac{T^2}{16\eta^2}\\ &+\Bigl(\frac{T}{8\eta^2}+\frac14\Bigr) \widehat\Xi_{T,\eta}(\mu,E)\biggr]. \end{aligned}

(33.9)

Consequently, for $0<T<1/8$ , $\eta=T^{1/3}$ , and $\eps\le T^{1/3}$ ,

\int_0^T\E\bigl[\bar e_t(E)\one_{\{t<\tau_\eta\}}\bigr]\dd t \le Te_0+C h_\mu\bigl(T^{4/3}+\widehat\Xi_{T,\eta}(\mu,E)\bigr).

(33.10)

This is a stopped refinement of Theorem 33.1, not a universal bound on the new interface.

Remark 33.1 (Features)

(a) The estimate holds for every balanced cut; near-minimality enters only through the size of $e_0$ . The hypothesis is on the measure — near-worstness — which is exactly the hypothesis a proof by contradiction is entitled to. (b) All randomness is compressed into the scalar functional $\Xi_T$ , and crucially $\Xi_T$ enters multiplied by $h_\mu$ : in the contradiction regime $h_\mu\approx\hstar_n$ is small, so the bootstrap tolerates covariance excursions of size up to $\sim1/\hstar_n$ — a genuine weakening of the operator-norm control warned about at (30.10). (c) A pathwise variant holds with constant probability: on $\{\tau_\eta>T\}\cap\{\sup_{t\le T}\lmax(A_t)\le1+\sigma\}\cap\{\sup_{t\le T}\mu_t^+(E)\le (1+\theta)\mu^+(E)\}$ — the last event of probability $\ge\theta/(1+\theta)$ by Doob’s maximal inequality for the nonnegative perimeter martingale — the same algebra gives $\sup_{t\le T}\bar e_t\le(1+\theta)e_0+h_\mu(\theta+\tfrac\eps2+\eta+\tfrac\sigma4)$ pathwise. Since the final step only requires events of probability $\ge\tfrac12$ , constant probability may suffice for a future stability argument needing pathwise near-minimality.

Evaluating $\Xi_T$ ¶

The polylogarithmic covariance estimates do beat it. The required input is the small-time operator-norm control imported in Chapter Small-time operator-norm control of the covariance (Assumption 26.1), discharged there on the published $c/\log^2 n$ window and, conditional on the version-1 input of Theorem 25.1, on the larger $c/\log n$ window (Corollary 26.2 and Corollary 26.3).

Corollary 33.1 (Polylog evaluation — Proved · agent review (gpt-6-astra, 2026-10-04) · cor:loglog · Correction)

Under Assumption 26.1, for $t_1(n)\le T\le1$ , $\Xi_T(\mu)\le C_1t_1(n)+\log(T/t_1(n))\le C(1+\log\log n)$ . Hence, when additionally $T<1/8$ and the near-worst parameter satisfies $\eps\le T^{1/3}$ , every such $\mu$ and every balanced cut $E$ satisfy, by (33.7),

\int_0^T\E\bigl[\bar e_t\,\one_{\{t<\tau_\eta\}}\bigr]\dd t \ \le\ Te_0+C\,h_\mu\bigl(T^{4/3}+1+\log\log n\bigr).

(33.11)

Proposition 33.1 (The relative-scale ceiling — Proved · agent review (gpt-6-astra, 2026-10-04) · prop:ceiling · Correction)

Fix $\kappa\in(0,1]$ and a universal $T_0>0$ satisfying

9(1+\kappa)T_0\le\frac12.

(33.12)

If $\Xi_{T_0}(\mu)\le\kappa T_0$ held for every isotropic log-concave $\mu$ , KLS would follow directly, with no geometric input. Consequently, a propagation estimate at relative scale — $\int_0^T\E\bar e_t\dd t\le\kappa h_\mu T$ with $\kappa$ small — cannot be expected from Theorem 33.1 alone: deriving it through (33.7) would require $\Xi_T\le c\kappa T$ at a sufficiently small universal time, an input already sufficient for KLS.

Remark 33.3 (The supply curve and the corridor)

Assembling this section and the previous one, the propagation estimates available at each scale are:

Scale of the excess bound	Scope	Source
Absolute, $\le C(1+e_0)T$	No hypothesis: all $\mu$ , all balanced $E$	Proposition 32.2.
Absolute with small constant, $\le Te_0+\kappa T$	Holds for near-worst $\mu$ whenever $h_\mu(1+\log\log n)\le c\kappa T$ ; conditional on Ass. Assumption 26.1	Corollary 33.1; the universal-time near-worst extension is Conjecture 29.4.
Relative, $\le\kappa\,h_\mu T$	Making the upper bound in (33.7) small term by term requires $\Xi_T\le c\kappa T$ , alongside control of $e_0$ and $T^{1/3}$ . If proved for every isotropic log-concave $\mu$ at a sufficiently small universal time, this covariance bound would imply KLS. A large upper bound gives no lower bound on the actual excess. Conjecture 29.4 asks instead for near-worst, $h_\mu$ -weighted control.	Proposition 33.1.

On the demand side, Proposition 25.2 shows the Stein-trace estimate cannot be satisfied slice-wise with absolute-scale excess. The corridor between what can be supplied and what must be demanded is the residual content of the approach, posed precisely in Chapter The fixed cut: remaining problems.

Remark 33.4 (Product measures: sanity check, and why the bootstrap is silent there)

For product $\mu$ (Chapter Model geometries: the Gaussian and product brackets), $A_t$ is diagonal, each entry a nonnegative supermartingale (drift $-A^2$ ), so $\Prob(\sup_{s\le t}A^{(i)}_s\ge\lambda)\le1/\lambda$ by the maximal inequality. A more careful small-time analysis of the $n$ independent variance processes — which we do not carry out — suggests $\Xi_T\asymp\log\log n$ for products of two-sided exponentials, matching Corollary 33.1 and consistent with the known observation that a single eigenvalue can reach order $\log n$ , saturating the cap (23.10) at $t\asymp1/\log n$ Chen, 2021Klartag & Lehec, 2022. For products, however, $h_\mu\asymp1\gg\hstar_n$ in any hypothetical bad regime, so the near-worstness hypothesis fails and the bootstrap is vacuous — correctly so: products cannot be counterexamples, and their excess propagation follows directly from $h_{\mu_t}\ge c\,\lmax(A_t)^{-1/2}$ by tensorization (Proposition 27.2).

Methodological constraints from this chapter¶

Both remarks below are methodological constraints on the bootstrap input, warnings rather than theorems; later chapters use them as heuristic barriers, never as a step in a proof.

Proof of the bootstrap comparison¶

The proof of Theorem 33.1, in the five steps outlined after its statement.

Proof 33.6 (Proof of Theorem 33.1)

Step 1. By the perimeter supermartingale (23.27) and Lemma 33.1, $\E[\mu_t^+(E)\one_{\{t<\tau_\eta\}}]\le\mu^+(E)=\tfrac{h_\mu}2+e_0$ .

Step 2. On $\{t<\tau_\eta\}$ , $\min(p_t,q_t)\ge\tfrac12-\eta$ . By Lemma 33.2 and the elementary bound $\lambda^{-1/2}\ge1-\tfrac12(\lambda-1)_+$ for $\lambda>0$ (trivial when the right side is negative),

\E\bigl[h_{\mu_t}\min(p_t,q_t)\one_{\{t<\tau_\eta\}}\bigr] \ \ge\ \hstar_n\Bigl(\frac12-\eta\Bigr)\Bigl(\Prob(\tau_\eta>t)-\frac{\E X_t}2\Bigr).

(33.14)

Step 3. Write

P=\Prob(\tau_\eta\le t),\qquad Y=\E X_t,\qquad a=\frac12-\eta,\qquad \rho=\frac{\hstar_n}{h_\mu}.

(33.15)

Near-worstness and the definition of $\hstar_n$ give

\frac1{1+\eps}\le\rho\le1, \qquad\text{hence}\qquad \rho\ge1-\eps.

(33.16)

If $1-P-Y/2\ge0$ , Steps 1–2 give

\begin{split} \E[\bar e_t\one_{\{t<\tau_\eta\}}]-e_0 &\le h_\mu\left\{\frac12-(1-\eps)a(1-P-Y/2)\right\}\\ &\le h_\mu\left(\eta+\frac P2+\frac Y4+\frac\eps2\right). \end{split}

(33.17)

Indeed, $\frac12-a(1-P)=\eta+aP\le\eta+P/2$ , the $Y$ contribution is at most $Y/4$ , and the $\eps$ contribution is at most $\eps/2$ .

If $1-P-Y/2<0$ , use only the nonnegativity of $h_{\mu_t}\min(p_t,q_t)\one_{\{t<\tau_\eta\}}$ in Step 1. The case assumption gives $P/2+Y/4>1/2$ , and therefore

\E[\bar e_t\one_{\{t<\tau_\eta\}}]-e_0 \le\frac{h_\mu}{2} <h_\mu\left(\eta+\frac P2+\frac Y4+\frac\eps2\right).

(33.18)

Thus (33.4) holds in both cases.

Step 4. The stopped process $M_u=p_{u\wedge\tau_\eta}-\tfrac12$ is a bounded continuous martingale. On $\{\tau_\eta\le t\}$ , continuity gives $\sup_{u\le t}|M_u|\ge\eta$ . Hence the $L^2$ maximal inequality, martingale isometry, (30.3), and $s_tr_t\le\tfrac14\lmax(A_t)\le\tfrac14(1+X_t)$ give

\begin{split} \Prob(\tau_\eta\le t) &\le\eta^{-2}\E M_t^2 =\eta^{-2}\E[p]_{t\wedge\tau_\eta}\\ &\le\frac1{4\eta^2}\int_0^t \E[\one_{\{s<\tau_\eta\}}(1+X_s)]\dd s\\ &\le\frac1{4\eta^2}\left(t+\int_0^t\E X_s\dd s\right). \end{split}

(33.19)

This proves (33.5).

Step 5. Integrate (33.4) over $[0,T]$ using $\int_0^T\Prob(\tau_\eta\le t)\dd t\le\tfrac1{4\eta^2}(\tfrac{T^2}2+T\Xi_T)$ .

References¶

Bobkov, S. G. (1999). Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures. Annals of Probability, 27(4), 1903–1921. 10.1214/aop/1022874820
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
Klartag, B. (2023). Logarithmic Bounds for Isoperimetry and Slices of Convex Sets. Ars Inveniendi Analytica, 2023(4), 1–17. 10.15781/jsjy-0b06
Chen, Y. (2021). An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture. Geometric and Functional Analysis, 31(1), 34–61. 10.1007/s00039-021-00558-4
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