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Distances between non-symmetric convex bodies: optimal bounds up to polylog

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Pith's one-line read This paper proves that any two n-dimensional convex bodies, with no symmetry assumption, have Banach-Mazur distance at most C n (log n)^α, optimal up to polylog factors.

desk verdict Real near-optimal Banach–Mazur result with two small repairable slips; send to review. read the letter →

arxiv 2510.20511 v3 pith:ZKFBL3DI submitted 2025-10-23 math.MG math.FAmath.PR

Pierre Bizeul , Boaz Klartag This is my paper · ORCID
classification math.MGmath.FAmath.PR MSC 52A2052A21
keywords Banach-Mazurdistanceconvexbodiesisotropicpositionstochasticlocalizationgaugeorderpartialcontainmentlog-concavemeasuresM-bound
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 shows that every pair of n-dimensional convex bodies — no symmetry assumed — are within a factor of C n (log n)^α of each other in the Banach-Mazur sense: after an affine change of coordinates and a translation, each scaled body contains the other. This improves the previous n^{4/3} guarantee and is optimal up to a polylogarithmic factor, matching what was already known for symmetric bodies. The proof works in the isotropic position and is driven by a new two-sided comparison between log-concave measures and Gaussian measures in the gauge order, i.e., uniformly over all norm-like functions. As a byproduct, the paper proves that a relaxed distance, where each body must contain only 99% of the other, grows only polylogarithmically, showing that the large Banach-Mazur distance is carried by small-volume outliers.

What carries the argument

Stochastic localization: starting from the uniform measure on a convex body, tilt the density by exp(θ·x - t|x|²/2) where θ_t is driven by Brownian motion, and track the barycenter a_t and covariance A_t of the tilted measure. The barycenter is a martingale with differential A_t dB_t, so the original random vector equals ∫ A_t dB_t in law. The argument hinges on a competition between the drift -A_t² dt and noise from third-moment tensors: for time T of order 1/(κ_n² log n), A_t stays between Id/2 and 2Id with high probability. Clipping A_t to that window yields a martingale v_t that is provably sandwiched between Gaussian expectations via two convexity arguments (one using an endpoint Gaussi

What would settle it

Run the stochastic localization SDE numerically from an isotropic log-concave measure in, say, n=1000 dimensions and watch the smallest eigenvalue of A_t: if it dips below 1/2 before time c/(κ_n² log n), the high-probability event that a_t = v_t is false, and the paper's lower Gaussian comparison loses its foundation.

Watch

Extended reading notes

Core claim

The paper's central claim is that the non-symmetric Banach-Mazur diameter of the space of n-dimensional convex bodies is n^{1+o(1)}: for any two compact convex sets K1,K2 with nonempty interior, d_BM(K1,K2) ≤ C n log^α n, with universal constants and α ≤ 4. Because there exist pairs of symmetric bodies with distance at least c n, this is best possible up to the polylog factor. The proof follows from a new 'isotropic M-bound': for an isotropic log-concave random vector X (the uniform measure on a body in isotropic position, meaning zero mean and identity covariance) and a standard Gaussian G, the expectations E||X|| and E||G|| agree up to the factor κ_n √log n, uniformly over all gauge functi

Load-bearing premise

The proof's engine is the claim that during stochastic localization the covariance matrix of the tilted measure remains between half and twice the identity for a short time; the upper half is proved in the appendix, while the lower half is only sketched as 'nearly identical', and if that lower bound fails the key equality between the barycenter and the clipped martingale collapses.

Editorial extensions

If this is right

  • The non-symmetric Banach-Mazur diameter is now known to within a polylog factor; the hardest pair of convex bodies is only n (log n)^α apart, matching the symmetric-case order n.
  • If the isoperimetric constant ψ_n (the KLS parameter) is eventually proved O(1), the same comparison sharpens to a √log n factor between log-concave and Gaussian expectations, and examples in the paper show that is optimal.
  • The isotropic position, after a random rotation, simultaneously realizes near-optimal containment for every pair of bodies, so it acts as an M-position without symmetry assumptions.
  • The partial-containment distance is polylogarithmic: any two isotropic bodies contain 99% of each other after scaling by about log^α n, showing the large Banach-Mazur distance is carried by small-volume outliers.
  • The new comparison has downstream uses in linear symplectic geometry and in bounding the first Dirichlet eigenvalue of the Laplacian.

Reading between the lines

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

  • Editorial: the exponent α≤4 is not claimed optimal; since the polylog enters through the current bound ψ_n ≤ C√log n, an improvement of that isoperimetric estimate would automatically improve the distances we can certify.
  • Editorial: the d_BM ~ n versus d_PC ~ log n gap suggests a clean dichotomy — bulk geometry is polylog-close, and the full-containment metric is inflated only by narrow tails; this could be tested by comparing random rotations of highly anisotropic bodies.
  • Editorial: the proof currently rests on a sketched lower-eigenvalue bound for the covariance process; turning that sketch into a complete argument is an immediate verification task for anyone using the M-bound.
  • Editorial: a natural continuation is to quantify how the partial-containment dilation grows as the required fraction β approaches 1; the paper's Markov argument gives β=99/100 only for a fixed universal constant, and a trade-off curve is not written down.
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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 / 4 minor

Summary. The paper studies the non-symmetric Banach–Mazur distance. Its main theorem (Theorem 1.1) asserts that any two convex bodies in R^n are at Banach–Mazur distance at most C n log^α n, improving Rudelson's n^{4/3} bound and matching Gluskin's lower bound up to polylog. The proof is via a random isotropic position (Theorem 1.2) and a new two-sided comparison, in the gauge order, between isotropic log-concave measures and Gaussian measures (Theorem 3.5 / Theorem 1.3). The right-hand comparison is attributed to Eldan and Lehec; the left-hand comparison is the paper's main new contribution, proved with stochastic localization and a covariance-process bound (Proposition 2.3). From the comparison and Chevet's inequality the authors derive M and M^* bounds in the isotropic position, an M M^* product O(polylog), a 99%-containment result for two isotropic bodies, and a polylog bound for the partial-containment distance d_PC. Applications to symplectic geometry and Dirichlet eigenvalues are also mentioned in the abstract, though the main body focuses on the geometric statements.

Significance. If the proof is repaired, this is a substantial advance: it essentially determines the diameter of the non-symmetric Banach–Mazur compactum up to polylogarithmic factors, matching the centrally-symmetric behavior, and it establishes a polylogarithmic partial-containment distance between arbitrary convex bodies. The new M-bound in the isotropic position is a natural and useful complement to E. Milman's M^*-bound, and the derivation of the geometric consequences from Theorem 3.5 and Chevet's inequality is clean and well organized. The stochastic-localization framework is presented in detail, with the main analytic proof being largely self-contained. The paper's central claims do not rely on fitted parameters or ad hoc entities; the new left-hand comparison is a genuine mathematical contribution. However, two load-bearing points need correction and completion before the argument is fully reliable.

major comments (2)
  1. [§3, Corollary 3.3 and proof of Theorem 3.5, Eqs. (41) and (50)] The left inequality in (41), EF(B_{t/2}) ≤ EF(v_t), is false as stated. Proposition 3.2 with r=1/4 yields EF(v_t) ≥ EF((1/2)B_t), not EF(B_{t/2}). The two processes have different covariances: B_{t/2} has covariance (t/2)I while (1/2)B_t has covariance (t/4)I, so for convex F the expectation with the larger covariance dominates; the asserted direction is the reverse. For n=1, A_t≡Id/2 and v_t=(1/2)B_t, we have E[B_{t/2}^2]=t/2 > t/4=E[v_t^2]. This false step is used at (50) in the proof of Theorem 3.5. The repair is local: replace B_{t/2} by (1/2)B_t in (41), and use (1/2)E||B_t|| ≤ E||v_t|| at (50). The subsequent derivation of (52) still goes through with an inconsequential change of the universal constant. The statement and proof must be corrected.
  2. [Appendix, Proposition 2.3, lower-eigenvalue half] The lower bound λ_min(A_t) ≥ 1/2 is load-bearing: Corollary 3.3's 'Moreover' part uses it to identify v_t with a_t, which is the basis for the new M-bound. The appendix explicitly says 'the argument is nearly identical, we only provide a sketch', and the final union bound is wrong as written. After bounding P(∃t≤T: λ_min(A_{t∧τ})≤1/2), the passage to λ_min(A_t) adds P(τ≥T); the correct term is P(τ≤T), because if no upper violation occurs before T, any lower violation is already a lower violation for the stopped process. Since Lemma A.4 only makes P(τ≤T) exponentially small, the displayed estimate ≤2exp(-c1/T) does not follow from the preceding lines. A complete proof of the lower-eigenvalue half is needed.
minor comments (4)
  1. [Proposition 2.3 statement] The probability is written as 'at least 1−Cexp(c/T)'; the exponent should be negative, i.e., '1−Cexp(−c/T)'.
  2. [Proof of Theorem 1.1] The sentence 'by Theorem 1.1, for λ=C√n log^2 n' should refer to Theorem 1.2.
  3. [Lemma 4.1 proof] In the chain 'E||G||_K ≤ Cκ_n √log n ∥X∥_K', the last term should be E||X||_K; as written it compares the expectation to a random variable.
  4. [Lemma A.2 proof] The identity ∑_i ⟨H_i^2 θ,θ⟩ = Tr[H_θ^2] is not self-evident for arbitrary symmetric matrices. It holds here because the H_i are coordinate slices of the third-moment tensor and the full symmetry of E[X_i X_j X_k] makes the two expressions equal. Adding one explanatory line would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main derivation is self-contained relative to external prior theorems, with no fitted parameter renamed as a prediction.

full rationale

The paper's central claim, Theorem 1.1, is derived from the new two-sided gauge comparison in Theorem 3.5, which is proved via stochastic localization (Corollary 3.3 and Proposition 2.3), with the appendix supplying the covariance-process bounds. The right-hand side of the comparison is imported from Eldan–Lehec, and the M*-bound is imported from E. Milman; these are prior published results with independent content, not definitions of the conclusion. The author-overlapping citations (Klartag's KLS bound, Klartag–Lehec stochastic-localization estimates, Bizeul's related work) are used as external theorems or methodological guidance, not as a substitute for the present proof; the present proof of Proposition 2.3 is provided in the appendix, and the paper does not fit any parameter to data and then call the fit a prediction. The apparent issue with the left-hand inequality in Corollary 3.3 (B_{t/2} versus (1/2)B_t) and the sketched lower-eigenvalue half of Proposition 2.3 are mathematical-correctness gaps in the written proof, not circularity: the stated inequalities are not equivalent to their inputs by construction. Even if those gaps require repair, the derivation chain does not reduce to a self-citation or to a definitional identity. Hence the appropriate circularity score is 0.

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

No data-fitting parameters appear: all constants are existential universal constants fixed by the proofs, not chosen ad hoc. The coefficient κ_n is a defined supremum over third moments, not a postulated physical or geometric object. The paper relies on several external deep theorems (KLS bound, Eldan–Lehec, Milman's M*) and on an internal proposition whose lower half is only sketched.

assumptions (7)
  • standard math Stochastic calculus: Itô's formula, martingale convergence, existence/uniqueness for the tilt SDE defining stochastic localization (Section 2, eq. (17)).
    Used throughout Section 2 and the appendix to define the tilt process, covariance dynamics, and the martingale a_t; standard background, not proved in the paper.
  • domain assumption KLS bound ψ_n ≤ C√logn (Klartag [25], eq. (5)).
    External deep theorem used to convert κ_n ≤ 2ψ_n into the explicit polylog exponents in Corollary 1.4 and Theorem 1.1. If this bound were false, the stated log exponents would fail.
  • domain assumption Eldan–Lehec upper bound: for an isotropic log-concave X and Gaussian G, E N(X) ≤ C ψ_n √logn E N(G) for seminorms N (Theorems 1.3 and 3.5, right-hand side).
    Cites [13] as a black box; the present paper explicitly says its contribution is the left-hand inequality.
  • domain assumption E. Milman's M* bound: for K in isotropic position, M*(K) ≤ C log² n √n (eq. (11), Corollary 1.4).
    External theorem [37] used in the proof of Theorem 1.2 and in the MM* product bound (12).
  • domain assumption Proposition 2.3: for T ≈ (κ_n² log n)^{-1}, Id/2 ≤ A_t ≤ 2Id for all t≤T with high probability.
    Stated as an internal result; the appendix gives a full proof for the upper bound but only a sketch for the lower bound. The central comparison of Corollary 3.3 depends on this proposition.
  • standard math Gaussian isoperimetric inequality and Sudakov-type comparison for suprema of Gaussian processes (used in Lemma 3.4 and Lemma 4.2).
    Standard tools cited to Ledoux and to Ledoux–Talagrand; not proved in the paper.
  • standard math Log-concave Lichnerowicz inequality: the tilted measure p_t has Poincaré constant at most 1/t (eq. (24)–(25)).
    Standard consequence of log-concavity; used to control the covariance process.

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Pith. "Pith review of Distances between non-symmetric convex bodies: optimal bounds up to polylog." pith.science (2026). https://pith.science/paper/ZKFBL3DI

@misc{pith2026251020511,
  author       = {Pith},
  title        = {Pith review of: Distances between non-symmetric convex bodies: optimal bounds up to polylog},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKFBL3DI}},
  note         = {Machine review of arXiv:2510.20511}
}
abstract

In this paper we determine, up to polylogarithmic factors, the diameter of the Banach--Mazur compactum of $n$-dimensional convex bodies without symmetry assumptions. We prove that for any convex bodies $K_1,K_2\subset \mathbb{R}^n$, \begin{equation} d_{BM}(K_1,K_2)\le Cn\log^\alpha(n+1), \label{eq_1624} \end{equation} for universal constants $C,\alpha>0$, improving an earlier bound of Rudelson. We also study the partial-containment distance $d_{PC}$, in which the Banach-Mazur requirement to contain the other body in its entirety is relaxed to $99\%$-containment. We prove that this relaxation leads to a very different behavior: \begin{equation} d_{PC}(K_1,K_2) \leq C \log^{\alpha} (n+1) \label{eq_1625} \end{equation} for all convex bodies $K_1,K_2 \subseteq \mathbb{R}^n$. This demonstrates that in high dimensions, any convex body is not too far from an affine image of any other convex body, when we look at the bulk of their mass. \medskip In the centrally-symmetric case, the optimal upper bound for the Banach-Mazur distance is obtained in the John position. In contrast, our proofs rely on the isotropic position. The analytic core of our argument is a two-sided comparison, in the gauge order, between isotropic log-concave measures and Gaussian measures. This yields a new isotropic $M$-bound that complements E. Milman's $M^*$-bound. We also provide applications to linear symplectic geometry and to the first Dirichlet eigenvalue of the Laplacian.

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Forward citations

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.