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Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read For every convex body of volume one, in every dimension, there exists a hyperplane whose intersection with the body has volume larger than a universal constant; equivalently, the isotropic constant is uniformly bounded across all…

desk verdict Likely-correct resolution of Bourgain's slicing problem, conditional on Guan's bound; the stress-test concern about Lemma 3.4 is a miscalculation and does not survive reading. read the letter →

arxiv 2412.15044 v1 pith:HYQDPZDH submitted 2024-12-19 math.MG math.FAmath.PR

Boaz Klartag , Joseph Lehec This is my paper · ORCID
classification math.MGmath.FAmath.PR MSC 52A2060D0546B06
keywords slicingproblemisotropicconstantconvexbodyhyperplanesectionstochasticlocalizationShannon-StaminequalityM-ellipsoidlog-concavemeasure
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims to settle the slicing problem, a long-standing question in convex geometry, in the affirmative. Its main theorem says that every convex body of volume one has a hyperplane section whose volume is larger than a universal constant that does not depend on the dimension. The equivalent formulation proved is a uniform bound on the isotropic constant of all convex bodies, a quantity that measures how far a body is from being Gaussian-like. The proof combines stochastic localization, a recently imported covariance bound, and a stability version of the Shannon-Stam inequality to force the entropy of any extremal body to be at least a constant times the dimension.

What carries the argument

The central object is the covariance process $(A_t)_{t \ge 0}$ generated by stochastic localization: starting from an isotropic log-concave measure, $A_t$ is the covariance of the tilted density $p_{t,\theta_t}$. Its evolution obeys $\frac{d}{dt} \mathbb{E} A_t = -\mathbb{E} A_t^2$, and the imported bound $\mathbb{E} \operatorname{Tr}[A_t^2] \le C n$ makes the expected trace decay at most linearly. With the substitution $r = t/(t+1)$ and $\Gamma_r = (1+t)A_t$, the process is related by $\mathbb{E}|v_r|^2 = J(\nu_r \|\gamma_r)$ to the Fisher information of the running law against a Gaussian, so integrating $r$ from 0 to 1 recovers the relative entropy $D(\mu \|\gamma_1)$. A stability estimate for the Shannon-Stam inequality turns the variance of $\Gamma_r$ into a lower bound on the same relative entropy, and an entropy jump bound $\delta_{KL}(\mu) \le 2m$ closes the argument.

What would settle it

Compute the quantity $\mathbb{E}\operatorname{Tr}[A_t^2]$ for the uniform measure on the simplex in high dimensions; if it ever exceeds $C n$ for a fixed universal $C$ and all $t$, the imported bound fails and the present proof collapses. More directly, any sequence of volume-one convex bodies whose maximal hyperplane-section volumes tend to zero would refute the theorem itself.

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Extended reading notes

Core claim

The paper proves Theorem 1.1: for any convex body $K \subset \mathbb{R}^n$ of volume one, there exists a hyperplane $H$ such that $\operatorname{Vol}_{n-1}(K \cap H) > c$, where $c > 0$ is universal. This is shown to follow from Theorem 1.2, $\sup_{n \ge 1} L_n < \infty$, where $L_n$ is the largest isotropic constant among $n$-dimensional convex bodies. The proof runs by contradiction at the extremal body: M-ellipsoid theory supplies a projection of the body onto a subspace of dimension $n/3$ whose isotropic constant is comparable to $L_n$, and stochastic localization generates a covariance process $A_t$ whose expected trace decays at most linearly thanks to the imported bound $\mathbb{E} \operatorname{Tr}[A_t^2] \le C n$. After a change of variables into the F\"ollmer drift, the same process controls the Fisher information along the heat flow, which integrates to the relative entropy with respect to the Gaussian law. A stability estimate for the Shannon-Stam inequality, combined with an entropy jump bound, then shows that the entropy of the projected measure is at least $-C m$; this forces $L_n \le C$ and completes the proof.

Load-bearing premise

The proof relies on the imported bound that, for an isotropic log-concave measure, the expected squared trace of the stochastic-localization covariance is at most a universal constant times the dimension; this lemma is not proved in the paper and, if it fails or needs extra hypotheses, the main theorem is unsupported.

Editorial extensions

If this is right

  • The slicing problem is settled: every convex body of volume one has a hyperplane section of volume at least a universal constant, in every dimension.
  • The isotropic constant $L_n$ is uniformly bounded across all dimensions, so every log-concave probability measure has isotropic constant within a universal multiplicative range of the Gaussian value.
  • The theorem strengthens M-ellipsoid theory: uniform lower bounds on hyperplane sections now hold without any dimensional factor.
  • The proof yields, in principle, an explicit -- though astronomically large -- universal constant for the slicing bound.
  • Conditionally, if the extremal body for $L_n$ is the simplex, the argument connects to Mahler's conjecture on volume products.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The theorem's correctness currently rests on an imported covariance bound that the paper does not prove; a self-contained verification of that bound would fully settle the argument.
  • The proof identifies the entropy jump bound as the only place where the dimension enters linearly, so refining the stability constants in the Shannon-Stam inequality would directly reduce the universal constant.
  • The mechanism may extend beyond log-concave measures: any family of measures with a covariance-process bound analogous to $\mathbb{E}\operatorname{Tr}[A_t^2] \le C n$ would inherit a slicing-type statement.
  • A natural next target is the strong slicing conjecture; the information-theoretic formulation points to equality cases in the entropy jump bound rather than to geometric symmetrization.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The paper claims to resolve Bourgain's slicing problem by proving sup_n L_n < ∞, subject to Guan's bound ETr[A_t^2] ≤ Cn for the stochastic localization covariance process. The proof selects an isotropic log-concave measure in a lower-dimensional subspace that nearly attains the maximal isotropic constant and whose expected covariance has a uniform lower bound, then closes the argument using the Eldan--Mikulincer stability estimate for the Shannon--Stam inequality together with an entropy bound of Ball--Nguyen. The internal chain from Guan's bound to the theorem is coherent, and I have verified the entropy-closing estimates, including the epsilon factor in Lemma 3.4.

Significance. If Guan's bound is correct, this is a landmark result: it provides the final step toward the affine-invariant hyperplane conjecture. The paper's own contribution is the new reduction from the covariance-process bound to a uniform isotropic constant, combining Milman's M-ellipsoid theory with the recent stability estimates. The derivations in Sections 2--4 are clear, and the proof is genuinely non-circular: Guan's bound is strictly weaker than the target statement. The principal caveat is the complete reliance on an unreviewed external preprint for the key estimate, which currently prevents the manuscript from being a self-contained proof of the slicing problem. I also examined the concern about the epsilon factor in Lemma 3.4; it does not land, because the matrix bound M ≤ 2ε^{-1}I yields M^{-1} ≥ (ε/2)I and hence exactly the claimed ε in the consequent, not ε².

major comments (2)
  1. [Section 2, Lemma 2.3] The central estimate ETr[A_t^2] ≤ Cn is imported verbatim from Guan's preprint arXiv:2412.09075 and is not proved or independently verified in this manuscript. This estimate is load-bearing: it enters at equation (28), where it yields the linear decay d/dt ETr[A_t] ≥ −Cn, and it drives Proposition 2.4 and the final entropy closing argument in Section 4. The footnote pointing to informal notes at a tinyurl is not a substitute for a proof. The manuscript should either include a complete proof of Lemma 2.3 (with appropriate attribution) or explicitly state in the abstract and introduction that the main theorem is conditional on this external result, pending independent verification. As written, the abstract's unconditional assertion is not supported by the body of the paper.
  2. [Abstract and Introduction] The paper's title and abstract state an unconditional resolution of Bourgain's slicing problem, yet the proof depends entirely on an unreviewed preprint posted six days earlier. This is a substantive accuracy issue: the reader is not told that the theorem is conditional unless they notice the footnote. The authors should make the conditional status explicit in the abstract and introduction, or include a proof of the external bound, before the paper can be accepted.
minor comments (3)
  1. [Section 2 and throughout] The notation '/greaterorsimilar' and '/lessorsimilar' (e.g., in Proposition 2.4 and Lemma 2.5) is nonstandard; the paper already defines ≲ and should use it consistently for both upper and lower bounds.
  2. [Section 2, Lemma 2.3 footnote] The footnote referencing an informal note via a tinyurl is not a stable scholarly reference. If the notes are needed for context, they should be made into a permanent appendix or replaced by a direct reference to Guan's preprint.
  3. [Section 3, Lemma 3.4] The consequence of Lemma 3.4 is correct, but the authors could make the derivation more transparent by explicitly stating that from Γ_r ≤ ε^{-1}I one has M := sqrt(((Γ^(1))^2+(Γ^(2))^2)/2) + (Γ^(1)+Γ^(2))/2 ≤ 2ε^{-1}I, so M^{-1} ≥ (ε/2)I; this intermediate step is what turns the displayed trace inequality into the ε-factor consequence.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proof is a forward deduction from external, process-level inputs (Guan's covariance bound, Eldan–Mikulincer stability, Ball–Nguyen entropy jump) and independent structural theorems, not from its own conclusion.

full rationale

The claimed derivation does not assume its conclusion. Guan's Lemma 2.3 is an external bound on the stochastic-localization covariance process (ETr[A_t^2] ≤ Cn) and is not stated in terms of sup L_n < ∞; the paper uses it as a hypothesis in a forward implication. The Eldan–Mikulincer stability estimate (Lemma 3.4), the Ball–Nguyen entropy bound (Lemma 3.5), and the Bourgain–Klartag–Milman extremal-body facts are parameter-free external theorems whose stated assumptions do not include the target inequality. The self-citations to [21], [22], [23], [26] support standard stochastic-localization identities, previous weaker bounds, or background facts, and do not encode the final constant bound. Proposition 2.4 constructs a measure with L_ν ≥ c L_n and a covariance lower bound; Section 4 then bounds L_ν from above using the imported inputs. No fitted parameter is renamed as a prediction, and no definitional equivalence ties the inputs to the output. The main external dependency, Guan's unproved-in-this-paper bound, is a verification limitation rather than a circular input; the possible ε-vs-ε² defect in the passage from Lemma 3.4 to (65), if real, is a correctness concern, not circularity.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on one unverified external input (Guan's Lemma 2.3) plus several established results from the published literature. There are no fitted parameters and no new postulated entities. The proof itself is a forward derivation from these ingredients.

assumptions (7)
  • domain assumption Guan's Lemma 2.3: for an isotropic log-concave probability measure in R^n, the stochastic localization covariance process satisfies E Tr[A_t^2] ≤ C n for all t > 0.
    Imported from arXiv:2412.09075 (Guan), a preprint posted 2024-12-11; not proved in this paper. The proof of Theorem 1.2 starts from this bound at (28), so the entire paper is conditional on it. The authors link to notes but do not reproduce the argument.
  • domain assumption The supremum L_n in (2) is attained by some convex body K (eq. (18)).
    Used to choose a maximizer K for the M-ellipsoid argument and to invoke [7, Corollary 3.5]. Stated without proof; standard compactness, but an unverified step.
  • standard math Milman's M-ellipsoid covering estimate (21): every convex body K is within covering number e^{Cn} of a volume-matched ellipsoid.
    Cited to Milman [30] and textbooks; used in Lemma 2.5 to lower-bound all (n-ℓ)-dimensional sections of K.
  • standard math [7, Corollary 3.5]: for a maximizer K, every k-dimensional central section satisfies Vol_k(K∩E)^{1/(n-k)} ≤ C.
    Previous result of Bourgain, Klartag, and Milman; used to derive the product bound (24) from the M-ellipsoid axes.
  • standard math Fradelizi's theorem: for centered K, max_x Vol_{n-ℓ}((x+K)∩F) ≤ C^n Vol_{n-ℓ}(K∩F).
    Cited to Fradelizi [14]; converts maximum sections to central sections in (26)-(27).
  • standard math Eldan-Mikulincer stability for Shannon-Stam (Lemma 3.4): δKL(μ) is lower-bounded by an integral of the squared deviation of Γ_r.
    Quoted from Eldan and Mikulincer [13], published in PTRF 2020; the quantitative lower bound in Lemma 3.4 is the engine of Section 4.
  • standard math Ball-Nguyen entropy jump: for isotropic log-concave μ, δKL(μ) ≤ 2n.
    Quoted from Ball and Nguyen [2], Studia Math 2012; provides the upper bound that closes the argument.

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Pith. "Pith review of Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound." pith.science (2026). https://pith.science/paper/HYQDPZDH

@misc{pith2026241215044,
  author       = {Pith},
  title        = {Pith review of: Affirmative Resolution of Bourgain's Slicing Problem using Guan's Bound},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HYQDPZDH}},
  note         = {Machine review of arXiv:2412.15044}
}
abstract

We provide the final step in the resolution of Bourgain's slicing problem in the affirmative. Thus we establish the following theorem: for any convex body $K \subseteq \mathbb{R}^n$ of volume one, there exists a hyperplane $H \subseteq \mathbb{R}^n$ such that $$ Vol_{n-1}(K \cap H) > c, $$ where $c > 0$ is a universal constant. Our proof combines Milman's theory of $M$-ellipsoids, stochastic localization with a recent bound by Guan, and stability estimates for the Shannon-Stam inequality by Eldan and Mikulincer.

Discussion (0). Continue with ORCID to comment.

Forward citations

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