arxiv: 2604.12692 · v1 · submitted 2026-04-14 · 🧮 math.FA · math.MG· math.PR

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Banach-Mazur distances and basis constants of isotropic log-concave random spaces

Apostolos Giannopoulos , Antonios Hmadi

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Pith reviewed 2026-05-10 14:15 UTC · model grok-4.3

classification 🧮 math.FA math.MGmath.PR

keywords Banach-Mazur distanceisotropic log-concave measuresrandom polytopesnormed spacesbasis constantGluskin's theoremBourgain slicing problem

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The pith

Random normed spaces from m isotropic log-concave samples in R^n have Banach-Mazur distance at least c n over ln(1 + m/n) with high probability.

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

The paper proves a lower bound for the Banach-Mazur distance between two independent random normed spaces built from the absolute convex hulls of m points each sampled from an isotropic log-concave measure on R^n. With high probability this distance is at least c n divided by the logarithm of (1 plus m over n). The bound is sharp in both dimension n and sample size m, and it reaches the largest possible order n precisely when m is roughly equal to n. The result extends the classical Gaussian case of Gluskin's theorem to the full class of isotropic log-concave measures and thereby supplies evidence that the extremal geometry of the Banach-Mazur compactum is universal across this broad family. As further consequences the authors obtain sharp estimates for the basis constants of these spaces and show that the spaces lie far from those possessing a one-unconditional basis.

Core claim

If x1,...,xm and y1,...,ym are independent samples from an isotropic log-concave probability measure on R^n, then the normed spaces X_Bm and Y_Am generated by their absolute convex hulls satisfy d_BM(X_Bm, Y_Am) >= c n / ln(1 + m/n) with high probability. This lower bound is sharp in both n and m and recovers the extremal order n when m is approximately n. The result extends Gluskin's theorem from the Gaussian setting to general isotropic log-concave measures, provides evidence for a universality phenomenon in the extremal geometry of the Banach-Mazur compactum, and yields sharp estimates for the basis constants of the random spaces together with the fact that these spaces are far from the 1

What carries the argument

The pair of random normed spaces X_Bm and Y_Am generated by the absolute convex hulls of two independent m-point samples from an isotropic log-concave measure; the Banach-Mazur distance between them is the quantity shown to be typically large.

If this is right

When m is comparable to n the typical distance attains the maximal order n.
The basis constants of these random spaces admit sharp estimates of the same order.
The spaces lie at large distance from the subclass of normed spaces that admit a 1-unconditional basis.
The same separation holds uniformly for every isotropic log-concave measure, not merely for the Gaussian.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result indicates that many other metric properties of random polytopal spaces may also be insensitive to the precise choice of isotropic log-concave measure.
Analogous lower bounds might hold for other natural distances on the space of normed spaces or for other random constructions built from log-concave measures.
Numerical checks for concrete measures such as the uniform distribution on the cube or the Euclidean ball could test the sharpness of the logarithmic factor in the denominator.

Load-bearing premise

The underlying probability measures are isotropic and log-concave, so that uniform tail bounds coming from recent progress on Bourgain's slicing problem apply to the geometry of the generated polytopes.

What would settle it

For some isotropic log-concave measure in dimension n and some m, exhibiting a positive-probability event in which the Banach-Mazur distance between the two corresponding random spaces falls below c n / ln(1 + m/n) would falsify the claimed lower bound.

read the original abstract

We study the Banach-Mazur distance between random normed spaces generated by centrally symmetric random polytopes associated with isotropic log-concave measures in $\mathbb{R}^n$. We show that, in a wide range of parameters, if $x_1,\dots,x_m$ and $y_1,\dots,y_m$ are independent samples from an isotropic log-concave probability measure on $\mathbb{R}^n$, then the corresponding normed spaces $X_{B_m}$ and $Y_{A_m}$ generated by their absolute convex hulls satisfy, with high probability, $$d_{{\rm BM}}(X_{B_m},Y_{A_m}) \geqslant \frac{cn}{\ln(1+m/n)},$$ which is sharp in both $n$ and $m$ and recovers the extremal order $n$ when $m \approx n$. Our results extend Gluskin's theorem from the Gaussian setting to general isotropic log-concave measures, providing evidence for a universality phenomenon in the extremal geometry of the Banach-Mazur compactum. In addition, we investigate operator-theoretic properties of the associated random spaces and, as consequences, we derive sharp estimates for their basis constant and show that these random spaces are far from the class of spaces with a $1$-unconditional basis. The proofs combine probabilistic and geometric methods with recent advances related to Bourgain's slicing problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Extends Gluskin's theorem to isotropic log-concave measures with a sharp lower bound on Banach-Mazur distance that holds uniformly.

read the letter

The main point of this paper is that it extends Gluskin's theorem to isotropic log-concave measures. They show that for two independent samples of m points each from such a measure in R^n, the random spaces generated by their convex hulls have Banach-Mazur distance at least c n over log(1 + m/n) with high probability. This bound is sharp in the parameters and gives the full order n when m is comparable to n. What stands out is how they handle the extension. Instead of relying on Gaussian-specific properties, they use recent uniform estimates from the slicing problem to get tail bounds that work for the whole class. The proof combines these with probabilistic estimates on random polytopes, volume ratios, and net arguments to separate the spaces. This gives evidence for some universality in the extremal geometry of the Banach-Mazur compactum. On top of that, they look at operator properties and get sharp bounds on the basis constant, plus show that these spaces are typically far from having a 1-unconditional basis. The soft spots are limited. The result depends on the current state of the isotropic constant bounds, which are not sharp yet, but the paper makes the dependence clear and doesn't claim more than what follows from known slicing results. The additional results on basis constants and unconditional bases follow from the same machinery without introducing new difficulties. The citation pattern looks standard for the area, pulling in the right prior work on Gluskin and slicing. This paper is for researchers in asymptotic geometric analysis, particularly those interested in random normed spaces and distances in the Banach-Mazur compactum. A reader who knows the Gaussian case will see the value in the generalization. It has enough new content and technical care to deserve a serious referee. I would recommend putting it through peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves a lower bound on the Banach-Mazur distance between random normed spaces X_{B_m} and Y_{A_m} generated by the absolute convex hulls of m independent samples from an isotropic log-concave measure on R^n: with high probability, d_BM(X_{B_m}, Y_{A_m}) ≥ c n / ln(1 + m/n). The bound is sharp in both n and m, recovers the extremal order n when m ≈ n, extends Gluskin's theorem from the Gaussian case, and yields consequences for basis constants and distance from spaces with 1-unconditional bases. Proofs combine volume-ratio and net arguments with uniform tail estimates derived from known bounds on Bourgain's slicing problem.

Significance. If the central claim holds, the result supplies concrete evidence for a universality phenomenon in the extremal geometry of the Banach-Mazur compactum by replacing Gaussian-specific concentration with measure-uniform tail bounds (L_K ≲ n^{1/4} polylog). The paper's strength is the direct, parameter-free application of established probabilistic-geometric tools to the full isotropic log-concave class, together with matching upper bounds already available in the literature for the Gaussian case. This yields a clean, sharp statement without hidden dependencies or fitted constants.

minor comments (3)

The abstract invokes 'recent advances related to Bourgain's slicing problem' without a specific citation; adding the reference (e.g., the source of the L_K bound used) in the introduction or preliminaries would improve traceability.
In the statement of the main theorem, the dependence of the high-probability event on the dimension n and sample size m could be made fully explicit (including the precise failure probability) to facilitate direct comparison with Gluskin's original result.
Notation for the random polytopes A_m and B_m is introduced in the abstract but would benefit from a short clarifying sentence in the introduction before the statement of the main result.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive and accurate assessment of our manuscript. The referee's summary correctly captures the main theorem, its sharpness in the parameters n and m, the extension of Gluskin's theorem, and the consequences for basis constants. We appreciate the recognition of the universality aspect for isotropic log-concave measures. As the recommendation is for minor revision and no specific technical issues were raised, we will incorporate any editorial suggestions in the revised version.

read point-by-point responses

Referee: The manuscript proves a lower bound on the Banach-Mazur distance between random normed spaces X_{B_m} and Y_{A_m} generated by the absolute convex hulls of m independent samples from an isotropic log-concave measure on R^n: with high probability, d_BM(X_{B_m}, Y_{A_m}) ≥ c n / ln(1 + m/n). The bound is sharp in both n and m, recovers the extremal order n when m ≈ n, extends Gluskin's theorem from the Gaussian case, and yields consequences for basis constants and distance from spaces with 1-unconditional bases. Proofs combine volume-ratio and net arguments with uniform tail estimates derived from known bounds on Bourgain's slicing problem.

Authors: We thank the referee for this precise encapsulation of the results. The proof strategy indeed relies on volume-ratio estimates, epsilon-net arguments, and uniform tail bounds that follow from the current best known estimates on Bourgain's slicing problem (specifically the L_K bound of order n^{1/4} polylog n). No changes are required here. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives the lower bound on d_BM via direct probabilistic tail estimates on random polytopes, applying known uniform volume-ratio and concentration bounds from external advances on Bourgain's slicing problem (L_K ≲ n^{1/4} polylog) to control the failure probability that two independent samples fail to separate the spaces. These inputs are independent of the target result, the extension of Gluskin's construction replaces only the Gaussian-specific concentration with the slicing-derived tails, and sharpness follows from matching upper bounds already in the literature rather than any self-referential fit or definition. No equation reduces the claimed inequality to a fitted parameter or prior self-citation by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard domain assumptions that the measures are isotropic and log-concave, together with the applicability of recent slicing-problem results; no new entities are postulated and the only free parameter is the implicit positive constant c.

free parameters (1)

c
Positive constant whose precise value is not computed; the inequality is stated to hold for some c>0.

axioms (2)

domain assumption The probability measures are isotropic and log-concave.
Invoked in the definition of the random polytopes and in the application of slicing results.
domain assumption Recent advances on Bourgain's slicing problem supply the required tail bounds.
Used to obtain the high-probability lower bound.

pith-pipeline@v0.9.0 · 5557 in / 1387 out tokens · 65239 ms · 2026-05-10T14:15:43.946378+00:00 · methodology

discussion (0)

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