The slicing conjecture via small ball estimates

Pierre Bizeul

REVIEW 4 major objections 4 minor 12 cited by

The slicing conjecture via small ball estimates

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper proves an optimal small-ball estimate for all isotropic log-concave random vectors and derives the slicing conjecture from it.

desk verdict A serious alternative proof of slicing via small-ball estimates; the main idea is sound but the final constant chase has a genuine n-dependence slip that must be fixed. read the letter →

arxiv 2501.06854 v1 pith:JWJ6XVDT submitted 2025-01-12 math.FA math.PR

Pierre Bizeul This is my paper · ORCID

classification math.FAmath.PR MSC 46B0952A2060E15

keywords smallballestimateslog-concavemeasuresisotropicvectorsslicingconjecturestochasticlocalizationM-ellipsoidconvexbodiesconstant

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 aims to establish a sharp small-ball estimate: for every isotropic log-concave random vector in $\mathbb{R}^n$ there is a universal constant $c_0>0$ such that the probability of landing in any Euclidean ball of squared radius $\varepsilon n$ is at most $\varepsilon^{c_0 n}$. Earlier estimates had suboptimal exponents or carried logarithmic factors, so the new bound removes the last dimension-dependent losses. From this estimate the paper derives the slicing conjecture, the statement that every volume-one convex body has a hyperplane section whose $(n-1)$-dimensional volume is bounded below by a universal constant. The route goes through the stochastic localization process, a covariance-trace lower bound imported as Lemma 8, and a conversion of small-ball control into a uniform bound on the isotropic constant via the theory of M-ellipsoids.

What carries the argument

The stochastic localization process is the central mechanism: starting from a log-concave measure $\mu$, it produces a random flow of $t$-strongly log-concave measures whose covariance satisfies $A_t \le t^{-1} I$, and along this flow the measure of any set shrinks at a rate controlled by the support diameter. The paper combines a covariance-trace lower bound (Lemma 8), which guarantees $E\operatorname{Tr}(A_{c_1})\ge c_1 n$ at a fixed time $c_1$, with a set-shrinkage estimate (Lemma 9) to show that a small Euclidean ball cannot retain more than $\varepsilon^{c_0 n}$ of the original mass. A second ingredient is the equivalence between small-ball estimates and the slicing bound, established through M-ellipsoid theory, which turns the probabilistic estimate into the uniform bound on the isotropic constant $L_n$.

What would settle it

Run the stochastic localization process on a concrete isotropic log-concave law, such as the uniform measure on a high-dimensional simplex or cube, and compute $E\operatorname{Tr}(A_{c_1})$ at the fixed time $c_1$; a value below $c_1 n$ would refute the proof's key input. Equivalently, find one isotropic log-concave vector, one center $y$, and one $\varepsilon<c_0$ with $P(\|X-y\|_2^2 \le \varepsilon n)>\varepsilon^{c_0 n}$ to falsify Theorem 1 directly.

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Extended reading notes

Core claim

The paper proves Theorem 1: for any isotropic log-concave vector $X$ in $\mathbb{R}^n$, there exists a universal constant $c_0>0$ such that for every $\varepsilon<c_0$ and every center $y\in\mathbb{R}^n$, $$P\left(\|X-y\|$_2^{2}$ \le \varepsilon n\right) \le \$varepsilon^{{c_0 n}}$.$$ This is a small-ball estimate with an exponent linear in the dimension, uniform over all centers and all isotropic log-concave measures. The proof reduces to vectors supported in a ball of diameter proportional to $\sqrt{n}$, runs the stochastic localization process up to a fixed universal time, and uses Lemma 8's lower bound on the expected trace of the covariance to prevent any small ball from retaining too much mass. Theorem 1 then yields the slicing conjecture, $\sup_n L_n < \infty$, by applying the estimate to the uniform measure on an extremal isotropic body, which lies in M-position, so the small-ball bound forces the isotropic constant to be bounded by a universal ratio of radii.

Load-bearing premise

The whole proof leans on an imported covariance-trace lower bound (Lemma 8): starting from an isotropic measure, the expected trace of the covariance at a fixed time $c_1$ must be at least $c_1 n$; if this fails, Theorem 1 and its slicing consequence do not follow.

Editorial extensions

If this is right

For every isotropic log-concave vector, concentration near any point is bounded by $\varepsilon^{c_0 n}$ with no dependence on the KLS constant or a logarithmic factor.
The slicing conjecture follows: $\sup_n L_n < \infty$, so every convex body of volume one has a hyperplane section of $(n-1)$-dimensional volume bounded below by a universal constant.
Theorem 1 can be bootstrapped to all $\varepsilon>0$, giving $P(\|X-y\|_2^2 \le \varepsilon^2 n) \le (C''\varepsilon)^n$ for a universal $C''>0$.
The proof provides an independent route to the slicing theorem, replacing stability estimates for the Shannon-Stam inequality with small-ball estimates.
The small-ball and slicing statements are quantitatively equivalent, so the same argument yields both directions of the bootstrap between them.

Reading between the lines

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

The argument leaves implicit that any future improvement of the universal constant $c_0$ in Theorem 1 would translate directly into a better uniform bound on the isotropic constant $L_n$, since the M-ellipsoid comparison in Section 3.2 is quantitative.
Because only the diameter and measure of the target set enter the shrinkage estimate, the same strategy may yield sharp small-ball bounds for other sets, such as slabs or halfspaces, with consequences for projection estimates for log-concave measures.
The uniform-over-centers form of Theorem 1 suggests a dimension-free occupancy statement: no ball of radius $\sqrt{\varepsilon n}$ can capture a fraction larger than $\varepsilon^{c_0 n}$ of an isotropic log-concave measure, regardless of where the ball is placed.
If Lemma 8's trace bound holds with an improved constant, the proof's method would give a quantitative version of the slicing conjecture with a fully explicit universal constant.

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Formalized claims in Lean

Claim #1: The paper proves Theorem 1: for any isotropic log-concave vector $X$ in $\mathbb{R}^n$, there exists a universal constant $c_0>0$ such that for every $\varepsilon<c_0$ and every center $y\in\mathbb{R}^n$, $$P\left(\|X-y\|$_2^{2}$ \le \varepsilon n\right) \le \$varepsilon^{{c_0 n}}$.$$ This is a small-ball estimate with an exponent linear in the dimension, uniform over all centers and all isotropic

/-- @claim 1 The paper proves Theorem 1: for any isotropic log-concave vector $X$ in $\mathbb{R}^n$, there exists a universal constant $c_0>0$ such that for every $\varepsilon<c_0$ and every center $y\in\mathbb{R}^n$, $$P\left(\|X-y\|$_2^{2}$ \le \varepsilon n\right) \le \$varepsilon^{{c_0 n}}$.$$ This is a small-ball estimate with an exponent linear in the dimension, uniform over all centers and all isotropic -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The paper is a legitimate alternative route to the slicing conjecture, not a repackaging. The new ingredient is the small-ball formulation: instead of Klartag–Lehec's stability estimates for Shannon–Stam, Bizeul goes through Dafnis–Paouris and proves the whole thing from a dimension-free small-ball bound. The structure is clean: reduce to bounded support, run stochastic localization, bound the shrinkage of a small ball, compare with an M-ellipsoid extremizer. I read the reduction in §2.1 and the shrinkage Lemma 9 carefully; both work. The paper is also honest about leaning on Guan's bound and on the Bourgain–Klartag–Milman M-position result, which are external inputs shared with the original resolution. That is dependence, not circularity.

The soft spot is exactly where the reader and stress-test found it: the final step of Theorem 1 in §3.1. The constant c0 is defined as min(c, c1^6/(256 e^{C1}), c c1^8 n / 512). The third term contains n, so c0 is not universal and the argument violates the quantifier in the statement. The other issue is the event E1: Lemma 9 with t = c1 and D^2 = C1 n gives a factor e^{C1 n c1/2}, not e^{C1 n/2}. As written, E1 is stronger than what Lemma 9 guarantees, so the claimed probability lower bound P(E1) >= 1 - c1^2/4 does not follow. Both are concrete and localized, and both look repairable—one can enlarge C1 to absorb c1 and take c0 over a minimum that is dimension-free. But as submitted, the proof of Theorem 1 is incomplete.

Everything else in the paper reads as competent. The M-ellipsoid argument in §3.2 is standard and fine once Theorem 1 is in hand. The writing is terse but clear. The reader's verdict of "conditional" is right: the core idea is credible and the skeleton is visible, but the constant chase needs repair before the theorem is actually proved.

Who gets value from this: convex geometers and people working in high-dimensional probability who want a small-ball-based proof of slicing, or who care about the optimal small-ball exponent. It deserves a serious referee, but the referee should be told to check the final step in §3.1 line by line. I would send it to review with a request for revision, not desk reject it.

Referee Report

4 major / 4 minor

Summary. This paper claims an alternative proof of Bourgain's slicing conjecture by establishing a small-ball estimate for isotropic log-concave measures. Theorem 1 asserts that for every isotropic log-concave vector in R^n there is a universal c0 such that P(||X-y||^2 <= epsilon n) <= epsilon^{c0 n} for all epsilon < c0 and all y. The proof reduces to bounded support, applies stochastic localization, uses Guan's lower bound on Tr(A_c1), combines Paouris small-ball estimates with a shrinkage lemma, and derives sup_n L_n < infinity from Theorem 1 via the Dafnis-Paouris equivalence and M-ellipsoid arguments. The manuscript is concise and the logical architecture is coherent, but the quantitative final step in Section 3.1 contains several constant errors that, as written, prevent the proof from establishing the claimed universal exponent.

Significance. If repaired, the paper would yield the optimal linear-in-n small-ball exponent for all isotropic log-concave measures and an alternative proof of Bourgain's slicing conjecture, a major recent result. The link between small-ball probabilities and the slicing constant is made explicit, and the reduction to bounded support in Section 2.1 is carefully presented. The paper is not fully self-contained: it imports Guan's bound (Lemma 8) and the Bourgain-Klartag-Milman M-position theorem, but this is acceptable if those external results are correct. No circularity is apparent: Theorem 1 is proved from external lemmas and Theorem C is derived afterward. The main obstacles are local quantitative errors in the final constant chase, which appear repairable without changing the overall strategy.

major comments (4)

[Section 3.1, final displayed definition of c0] The definition c0 = min(c, c1^6/(256e^{C1}), c c1^8 n/512) is not admissible for Theorem 1 because the third entry depends on the dimension n. Moreover, to deduce mu(S_epsilon) <= epsilon^{c0 n} from the proved bound mu(S_epsilon) <= epsilon^{c c1^8 n/512}, one needs c0 <= c c1^8/512, not c0 <= c c1^8 n/512; for n>1 the displayed choice gives c0 n > c c1^8 n/512, so the claimed right-hand side is smaller than the proved bound. The proof must redefine c0 as a minimum of dimension-free constants.
[Section 3.1, event E1 and Lemma 9] The event E1 is written as {mu(S_epsilon) <= e^{C1 n/2} mu_t(S_epsilon)^{1/lambda}}, but Lemma 9 with D^2 = C1 n and t = c1 gives e^{C1 c1 n/2}, not e^{C1 n/2}. Unless c1 = 2, the event as stated is stronger than the lemma guarantees, so the asserted probability bound P(E1) >= 1 - c1^2/4 does not follow. The factor c1 must be absorbed into C1 or into the threshold for epsilon.
[Sections 2.1 and 3.1, support diameter] The reduction in Section 2.1 concludes that X3 is supported in B(0, 2*sqrt(2) C0 sqrt(n)), so its diameter is at most 4*sqrt(2) C0 sqrt(n), i.e. D^2 <= 32 C0^2 n. The text instead declares a support diameter sqrt(C1 n) with C1 = 8 C0^2, which is the radius squared, not the diameter squared. Since Lemma 9 and inequality (15) depend on the actual diameter D, the constant C1 used in Section 3.1 is too small by a factor of 4 in the worst case and must be enlarged.
[Section 3.1, application of Lemma 4 to mu_t] When Lemma 4 is applied to mu_t on the event E0, the Paouris parameter is epsilon' = epsilon n / Tr(A_t), and E0 only gives epsilon' <= 2 epsilon / c1. The displayed threshold epsilon <= c1^6/(256 e^{C1}) wedge c does not imply epsilon' < c, the threshold required by Theorem A, unless an additional condition such as epsilon <= c c1/2 is imposed. This is part of the same final constant chase and should be fixed when c0 is redefined.

minor comments (4)

[Lemma 5] The phrase 'There is a constant a constant C0' contains a duplicated word; it should read 'There is a constant C0'.
[Lemma 6] In the proof, the convex function g should be |x dot theta|^p rather than (x dot theta)^p, since for odd p the function t -> t^p is not convex on all of R.
[Introduction] The name 'Danis' in 'It was first observed by Danis and Paouris' is a typo for 'Dafnis'.
[Section 2.1] The notation for r and epsilon changes between the reduction and Section 3.1: the reduction sets r = epsilon sqrt(n), while Section 3.1 uses S_epsilon = B(0, sqrt(epsilon n)). The constants in the final theorem are unaffected, but the two epsilons should be distinguished to avoid confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning detected; the proof is an independent derivation from external lemmas, though it contains a quantifier error in the final constant as written.

full rationale

The paper's derivation chain is not circular. The main result, Theorem 1, is proved from stochastic localization, Guan's lower trace bound (Lemma 8), Paouris's small ball estimate, and a set-shrinkage lemma (Lemma 9). None of these inputs assumes the target small-ball estimate or the slicing conjecture. Theorem C is then derived from Theorem 1 via the Dafnis-Paouris equivalence and Milman's M-ellipsoid theory, and the direction used is precisely the one that goes from a small-ball estimate to a bound on the isotropic constant, not the reverse. The proof of Theorem C does not invoke Theorem C as an assumption. The reliance on Guan's bound is a dependency on an external result, but that is not circularity: Guan's bound is a covariance-trace estimate obtained independently, and the paper does not assume the conclusion of the slicing conjecture anywhere. The reduction in Section 2.1 to bounded support and to the case y = 0 is standard and does not define the target quantity in terms of itself. The proof of Theorem 1 uses Lemma 4 and Lemma 6 to convert strong log-concavity into subgaussianity, and these lemmas are not fitted to the desired small-ball probability. There is no fitted parameter that is later renamed as a prediction, and no self-citation is load-bearing; citations to Klartag-Lehec, Guan, and Dafnis-Paouris are to independent works. The only notable issue is a technical completeness problem in Section 3.1: the constant c0 is defined as the minimum of three quantities, the last being c c1^8 n / 512, which depends on n and therefore violates the dimension-free universal constant claim in Theorem 1. This is a correctness defect, not circularity, because it is a local quantifier error that can be repaired by taking an n-independent lower bound, not a step where the target result is assumed. Similarly, the exponent in the definition of E1 appears to omit a factor of c1 relative to Lemma 9; this is another local quantitative gap but does not make the derivation circular. Overall, the paper is self-contained relative to its cited external ingredients, and no circular step is evident.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard high-dimensional geometry background plus one recent external result, Guan's bound. No free parameters are fitted to data and no new entities are postulated. The proof is not formally verified.

assumptions (6)

domain assumption Paouris Theorem A: small-ball estimate for b-subgaussian log-concave vectors
Used in Lemma 4 to bound the probability for the localized strongly log-concave measure.
domain assumption Guan's bound: E Tr(A_{c1}) >= c1 n for an isotropic starting measure
External unproved input, load-bearing for Theorem 1; stated as Lemma 8.
domain assumption Log-concave Lichnerowicz inequality: A_t <= (1/t) I
Used in Lemma 10 and in the bounds on E Tr(A_{c1}).
domain assumption Dafnis-Paouris equivalence and Milman M-ellipsoid theory
Bridges small-ball estimates to the slicing constant in Section 3.2.
standard math Hargé convex/log-concave correlation inequality
Used in Lemma 6 to show that strongly log-concave vectors are subgaussian.
domain assumption Bourgain-Klartag-Milman theorem that extremizers of L_n are in M-position
Used in Section 3.2 to compare volumes and extract the bound on L_n.

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Cite this review

Pith. "Pith review of The slicing conjecture via small ball estimates." pith.science (2026). https://pith.science/paper/JWJ6XVDT

@misc{pith2026250106854,
  author       = {Pith},
  title        = {Pith review of: The slicing conjecture via small ball estimates},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JWJ6XVDT}},
  note         = {Machine review of arXiv:2501.06854}
}

read the original abstract

Bourgain's slicing conjecture was recently resolved by Joseph Lehec and Bo'az Klartag. We present an alternative proof by establishing small ball probability estimates for isotropic log-concave measures. Our approach relies on the stochastic localization process and Guan's bound, techniques also used by Klartag and Lehec. The link between small ball probabilities and the slicing conjecture was first observed by Dafnis and Paouris and is established through Milman's theory of M-ellipsoids.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [1]

Geometry of Banach spaces and harmonic an alysis

Jean Bourgain. Geometry of Banach spaces and harmonic an alysis. In Proceedings of the International Congress of Mathematicians , volume 1, page 2. Citeseer, 1986

work page 1986

[2] [2]

On high dimensional maximal functions as sociated to convex bodies

Jean Bourgain. On high dimensional maximal functions as sociated to convex bodies. Amer- ican Journal of Mathematics , 108(6):1467–1476, 1986

work page 1986

[3] [3]

Symmet rization and isotropic constants of convex bodies

Jean Bourgain, Bo’az Klartag, and Vitali Milman. Symmet rization and isotropic constants of convex bodies. In Geometric Aspects of Functional Analysis: Israel Seminar 2 002-2003, pages 101–115. Springer, 2004

work page 2003

[4] [4]

Small ball probabili ty estimates, ψ 2-behavior and the hyperplane conjecture

Nikos Dafnis and Grigoris Paouris. Small ball probabili ty estimates, ψ 2-behavior and the hyperplane conjecture. Journal of Functional Analysis , 258(6):1933–1964, 2010

work page 1933

[5] [5]

Thin shell implies spectral gap up to polylo g via a stochastic localization scheme

Ronen Eldan. Thin shell implies spectral gap up to polylo g via a stochastic localization scheme. Geometric and Functional Analysis , 23(2):532–569, 2013

work page 2013

[6] [6]

A note on Bourgain's slicing problem

Qingyang Guan. A note on Bourgain’s slicing problem. arXiv preprint arXiv:2412.09075, 2024

work page Pith review arXiv 2024

[7] [7]

A convex/log-concave correlation inequ ality for Gaussian measure and an application to abstract Wiener spaces

Gilles Harg´ e. A convex/log-concave correlation inequ ality for Gaussian measure and an application to abstract Wiener spaces. Probability theory and related ﬁelds , 130:415–440, 2004

work page 2004

[8] [8]

Discrete & Computational Geometry, 13:541–559, 1995

Ravi Kannan, L´ aszl´ o Lov´ asz, and Mikl´ os Simonovits.Isoperimetric problems for convex bodies and a localization lemma. Discrete & Computational Geometry, 13:541–559, 1995

work page 1995