Central diagonal sections of Gaussian cubes

Bernardo Gonz\'alez Merino; Ferenc Fodor

REVIEW 2 major objections 5 minor 4 references

The paper proves that for every b>0, the Gaussian-weighted area of the cube's central diagonal section tends to 2√(b/π) (1 − 4 e^{-b}√b/(2√π erf(√b)))^{-1/2} as n→∞.

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2026-08-04 00:21 UTC pith:QRTYHFJM

load-bearing objection New limit formula for Gaussian diagonal cube sections that is likely correct but whose written proof has a scaling error at its core. the 2 major comments →

arxiv 2511.01504 v3 pith:QRTYHFJM submitted 2025-11-03 math.MG math.FAmath.PR

Central diagonal sections of Gaussian cubes

Ferenc Fodor , Bernardo Gonz\'alez Merino This is my paper

classification math.MG math.FAmath.PR MSC 52A38

keywords Gaussian cubecentral diagonal sectionhyperplane sectionscube slicingerror functiongenerating functionasymptotic geometryproduct measure

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

For a cube carrying a Gaussian-type density e^{-b||x||^2} rather than uniform measure, the paper determines the exact limit, as the dimension n grows, of the induced (n−1)-dimensional measure of the hyperplane section through the origin orthogonal to the main diagonal. The limit is an explicit closed-form function of the parameter b, given by 2√(b/π) (1 − 4 e^{-b}√b/(2√π erf(√b)))^{-1/2}. This extends the classical Lebesgue-measure limit √(6/π), which is recovered in the limit b→0+. The result provides a first rigorous asymptotic benchmark for the still-open question of which hyperplane sections of Gaussian cubes are maximal, and the paper conjectures—with numerical support for small b—that the section measures increase monotonically toward this limit.

Core claim

The central discovery is Theorem 1.1: for every fixed b>0, the Gaussian-type section measure A(a,γ_n[b]) of the cube [-1,1]^n through the origin and orthogonal to the main diagonal tends to 2√(b/π) (1 − 4 e^{-b}√b/(2√π erf(√b)))^{-1/2} as n→∞. The proof starts from an integral representation of A(a,γ_n[b]), evaluates the cosine-Gaussian integral ∫_0^1 cos(rs)e^{-bs^2}ds in closed form via the error function, expands the resulting nth power in powers of r^2/n using a Taylor expansion of the error-function combination around √b, and then integrates term by term against e^{-r^2/4b}. The surviving infinite series is summed by the generating function ∑_{k≥1} a^k binom(2k−1,k) = ½(1/√(1−4a)−1), an

What carries the argument

The central identity is Lemma 2.1, which evaluates f_b(r)=∫_0^1 cos(rs)e^{-bs^2}ds as (√π/(4√b)) e^{-r^2/4b} [erf(√b − ri/(2√b)) + erf(√b + ri/(2√b))]. This turns the product over coordinates in the section formula into a single nth power of an error-function combination. The proof then Taylor-expands that combination around √b, yielding a power series in r^2/n; Gaussian moment integrals ∫_0^∞ e^{-r^2/4b} r^{2k} dr evaluate each term, and the series is recognized as the generating function ∑_{k≥1} a^k binom(2k−1,k) = ½(1/√(1−4a)−1). A monotonicity property of g(b)=e^{-b}√b/(2√π erf(√b)), with g(0+)=1/4, keeps the argument of the square root positive.

Load-bearing premise

The proof assumes that, after expanding the nth power in the Taylor series of the error-function combination, one may let n→∞ inside the integral and integrate the resulting infinite series term by term; no dominated-convergence or remainder estimate is supplied, so this interchange is the load-bearing step whose failure would invalidate the limit.

What would settle it

Numerically evaluate A(a,γ_n[b]) via the integral representation for a fixed b (e.g., 0.25) and n = 10^4, 10^5, 10^6; if the sequence does not approach the right-hand side of (1.2), the claimed limit is false. Alternatively, test the Taylor expansion's higher-order terms directly: if the neglected powers of r^2/n make a non-vanishing contribution to the integral as n→∞, the interchange used in the proof fails.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The b→0 limit of the formula recovers the classical Lebesgue limit √(6/π), so the Gaussian result extends the classic cube-slicing asymptotic.
If the paper's conjecture that A(a,γ_n[b]) is monotonically increasing in n (for n≥3) holds, the limit expression serves as an explicit upper bound for every central diagonal Gaussian section.
Numerical computations reported for b=0.1 and b=0.25 show the section measures approaching the limit from below across 2≤n≤50, consistent with the conjectured monotonicity.
The proof isolates a parameter g(b)=e^{-b}√b/(2√π erf(√b)) that is shown to lie in (0,1/4) and that governs the multiplicative correction to the leading 2√(b/π) term.
The paper describes the result as a first step toward determining which hyperplane sections of Gaussian cubes are maximal, a problem that remains open.

Where Pith is reading between the lines

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

The same Taylor-expansion-plus-Gaussian-moments strategy could yield asymptotic limits for sections in directions near the diagonal, or for coordinate-dependent weights e^{-b_j s_j^2}, where the integral representation still has a product structure.
The appearance of the generating function for binom(2k−1,k) suggests a possible probabilistic reading: the limit may equal an expectation under a discrete distribution with those weights, potentially connecting cube slice asymptotics to random-walk or branching-process quantities.
A testable extension would be to use the first terms of the expansion to derive non-asymptotic bounds on |A(a,γ_n[b]) − limit|; examining the signs of the contribution terms could either support or refute the conjectured monotone increase.
Because the delicate step is the interchange of limit and integral, a direct numerical evaluation of the integral representation for very large n (say 10^5) and a fixed b would quickly reveal whether the asymptotic formula is plausible.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

New limit formula for Gaussian diagonal cube sections that is likely correct but whose written proof has a scaling error at its core.

read the letter

Quick note on arXiv:2511.01504. The result is genuinely new: a b-dependent limit for Gaussian-weighted diagonal sections of the cube, extending Hensley's √(6/π). The formula in Theorem 1.1 looks right — an independent Laplace evaluation of the intended integral gives exactly that limit, and the b→0 limit recovers the Lebesgue case. So the mathematical contribution is real.

The problem is the written proof. Equation (2.4) does not follow from (1.1) as printed. With the cosine argument r√n in (1.1), Lemma 2.1 gives an integrand with e^{-n r^2/(4b)} and erf arguments of order r√n, not the e^{-r^2/(4b)} and r/(2√(nb)) in (2.4). The Taylor expansion and term-by-term integration that follow are then applied to a different integral. This is probably a typo — the intended formula almost certainly uses cos(rs/√n) — but as written the proof has a false displayed identity at its core.

Beyond that, the interchange of n→∞ with the integral and the truncation of the n-th power to the quadratic term are asserted without the necessary uniform estimates. These are standard Laplace-type steps, but they need to be written out. There is also a factor-of-2 mismatch between the abstract and Theorem 1.1, and a few sign errors in the higher-order Taylor coefficients that don't affect the limit.

The literature review is solid, the self-citations are used appropriately, and the conjecture about monotonicity is clearly labeled. The paper is short and the statement is clean.

Who should read this? People working on cube slicing, Gaussian sections, or asymptotic geometric analysis. It's a compact note with a likely-correct new limit, but the proof as it stands is not rigorous. I'd send it to a serious referee rather than desk-reject, because the result is important enough and the gaps are fixable. The authors need to correct the scaling, supply the missing estimates, and reconcile the abstract with the theorem. After that, it should be a solid publication.

I'd bring it to the reading group — the mismatch between (1.1) and (2.4) makes for a good exercise in checking asymptotic derivations.

Referee Report

2 major / 5 minor

Summary. The paper studies the Gaussian-type probability measure on the cube C^n=[-1,1]^n whose density is proportional to e^{-b||x||^2}, and the induced measure of central hyperplane sections. The main claim, Theorem 1.1, is that for every b>0, the measure of the section orthogonal to the main diagonal satisfies lim_{n→∞} A(a, γ_n[b]) = 2 sqrt(b/π) (1 - 4g(b))^{-1/2}, where g(b)=e^{-b} sqrt(b)/(2 sqrt(π) erf(sqrt(b))), recovering Hensley's sqrt(6/π) as b→0+. The proof starts from the König–Koldobsky integral formula (1.1), evaluates the relevant cosine transform in closed form (Lemma 2.1), proves a monotonicity property of g (Lemma 2.2), and then attempts a Taylor-expansion/series evaluation. The paper also includes a conjecture that A(a, γ_n[b]) is increasing in n for n≥3.

Significance. If the main theorem is correct, it is a natural and explicit extension of Hensley's classical cube-slicing result to Gaussian weights, and it would be a useful step toward understanding extremal Gaussian sections. The closed-form identity in Lemma 2.1 and the monotonicity lemma for g are correct and clean contributions. However, the written proof of Theorem 1.1 contains a serious scaling error at equation (2.4): the displayed integral is not what follows from (1.1) and Lemma 2.1. As a result, the subsequent asymptotic manipulations are applied to a different integral, so the theorem is not established by the submitted argument. The final formula may well be correct, and the paper's main idea is salvageable, but the proof requires substantial revision.

major comments (2)

[§2, Eq. (2.4)] Equation (2.4) does not follow from (1.1) and Lemma 2.1. Substituting Lemma 2.1 at argument r√n gives numerator (√π/(4√b)) e^{-n r^2/(4b)} [erf(√b - r√n/(2√b) i) + erf(√b + r√n/(2√b) i)]. Dividing by the normalization ∫_0^1 e^{-b s^2} ds = √π erf(√b)/(2√b) gives an integrand factor [1/(2 erf(√b))] e^{-n r^2/(4b)} [ ... ]. Hence the exponential in (2.4) should be e^{-n r^2/(4b)}, and the erf arguments have imaginary part r√n/(2√b), not r/(2√(nb)). The subsequent Taylor expansion in c=r/(2√(nb)) and all series manipulations are therefore applied to an incorrect integrand. A correct proof needs a Laplace-type analysis of the true integral, together with control of the remainder in the n-th power.
[§2, proof of Theorem 1.1 (after Eq. (2.4))] Independently of the scaling error, the limiting argument is not justified. The proof expands the n-th power, integrates term by term, and passes to n→∞ inside the integral without a dominated-convergence or uniform-integrability argument. The sentence 'Above we have already partly applied limit computations' does not supply the required justification. The passage from the finite binomial sum to the infinite Catalan series also needs justification, as does the interchange of the limit with the integral in the final evaluation. These gaps are load-bearing and must be closed before the proof is rigorous.

minor comments (5)

[Title/Abstract] The title and abstract use 'Gaussiann-cubes' with no space; it should read 'Gaussian n-cubes'.
[References] The references [AGBC] and [DLLCT] are missing publication years; please complete them before final submission.
[§3 / References] The generating-function identity Σ_{k≥1} a^k C(2k-1,k) = 1/2 ((1-4a)^{-1/2}-1) is cited to a Mathematics Stack Exchange answer [Som21]. A standard textbook reference would be more appropriate.
[§2, proof of Theorem 1.1] If the proof is repaired, the phrase 'Above we have already partly applied limit computations' should be replaced by an explicit convergence argument; the current wording obscures the main technical step.
[Figure 1] The figure would be more informative if the limiting value from (1.2) were plotted as a horizontal line; this is a presentation suggestion only.

Circularity Check

0 steps flagged

No circularity found: Theorem 1.1 is derived from the external KK13 integral formula and internal calculus; self-citations are contextual.

full rationale

The main theorem is obtained from the König–Koldobsky formula (1.1), an external result, by substituting the closed form of Lemma 2.1 (proven in the paper via an ODE) and asymptotically evaluating the resulting integral. The target limit is not an input anywhere: no parameter is fitted to the limit, no definition is given in terms of the result, and the only self-citations ([BFGM21]) are used for context or for a conjecture, not in the proof of Theorem 1.1. Although the asymptotic step after (2.4) may contain a scaling issue (a correctness question), that does not constitute circularity, since the claimed conclusion is not assumed. The series identity is external, and Lemma 2.2 is proven internally. Thus the derivation chain does not reduce to its inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The paper introduces no free parameters beyond the fixed b of the Gaussian density. The argument rests on the König–Koldobsky integral representation, standard error-function and Catalan identities, and an unproved limit interchange in the asymptotic step. No new entities are postulated.

axioms (4)

domain assumption The König–Koldobsky formula (1.1) correctly expresses A(a, γ_n[b]) as an integral over r
The paper starts from this cited result ([KK13, Prop. 2.1]) with a correction factor of 2; it is not proved in the paper. It is the exact representation of the induced section measure.
standard math The generating function identity Σ_{k≥0} binom(2k,k) a^k = 1/√(1-4a) holds
Used to sum the Catalan series; cited to [Som21]. This is a standard analytic-combinatorics identity, used with |a| < 1/4.
ad hoc to paper The limit n→∞ can be interchanged with the integral and the nth-power expansion can be truncated to the quadratic term
The paper asserts this implicitly ('Above we have already partly applied limit computations') without a dominated-convergence or remainder estimate. This is the main gap in the proof; it is load-bearing for the derivation as written.
standard math Standard properties of the error function erf, including the conjugate symmetry erf(conj z) = conj(erf z) and the Taylor expansion around √b, are valid
Used in Lemma 2.1 and in the expansion of erf(√b ± ic) in the proof of Theorem 1.1.

pith-pipeline@v1.3.0-alltime-deepseek · 7776 in / 40712 out tokens · 335290 ms · 2026-08-04T00:21:49.993772+00:00 · methodology

0 comments

read the original abstract

The investigation of the volume, surface area, and other geometric properties of sections of convex bodies, and in particular cubes, has a long history and a rich literature. However, much less is known when the cube has a volume distribution that is different from the Lebesgue measure; for example, a Gaussian density. We study the probability densities in the standard cube $B^n_\infty=[-1,1]^n$ of $\mathbb R^n$ generated by $e^{-b\|x\|^2}$, $b> 0$. We prove that the limit of the induced Gaussian-type volume of hyperplane sections of $B^n_\infty$ through the origin and orthogonal to a main diagonal is \[ \sqrt{\frac b\pi}\left (1-4\frac{e^{-b}\sqrt{b}}{2\sqrt{\pi}\mathrm{erf}(\sqrt{b})}\right)^{-\frac12}, \] as $n\to\infty$. This extends the well-known result of Hensley (1979) for the Lebesgue measure and continues the investigations initiated by Barthe, Gu\'edon, Mendelson, Naor (2005), Zvavitch (2008), and K\"onig, Koldobski (2013).

Figures

Figures reproduced from arXiv: 2511.01504 by Bernardo Gonz\'alez Merino, Ferenc Fodor.

**Figure 1.** Figure 1: The values of A(a, γn[b]) for b = 0.1 and b = 0.25 and 2 ≤ n ≤ 50. Funding sources F. Fodor’s research was supported by NKFIH project no. 150151, which has been implemented with the support provided by the Ministry of Culture and Innovation of Hungary from the National Research, Development and Innovation Fund, financed under the ADVANCED 24 funding scheme. B. Gonzalez Merino was partially supported by MI… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

4 extracted references · 1 canonical work pages

[1]

Abel,The number of gridpoints on hyperplane sections of thed-dimensional cube, Proc

[Abe18] U. Abel,The number of gridpoints on hyperplane sections of thed-dimensional cube, Proc. Amer. Math. Soc.146(2018), no. 12, 52495355. [Ali08] I. Aliev,Siegel’s Lemma and Sum-Distinct Sets, Discr. Comput. Geom.39(2008), no. 3, 59–66. [Ali21] I. Aliev,On the volume of hyperplane sections of ad-cube, Acta Math. Hungar.163 (2021), 547–551. [AGBC] D. Al...

arXiv 2018
[2]

K¨ onig and A

[KK11] H. K¨ onig and A. Koldobsky,Volumes of low-dimensional slabs and sections in the cube, Adv. Appl. Math.47(2011), no. 4, 894–907, DOI 10.1016/j.aam.2011.05.001. [KK13] H. K¨ onig and A. Koldobsky,On the maximal measure of sections of then-cube, Geo- metric analysis, mathematical relativity, and nonlinear partial differential equations, Contemp. Math...

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P´ olya,Berechnung eines bestimmten Integrals., Math

[P´ ol13] G. P´ olya,Berechnung eines bestimmten Integrals., Math. Ann.74(1913), 204–212. [Pou23a] L. Pournin,Local extrema for hypercube sections, Journal d’Analyse Math´ ematique 152(2023), no. 2, 557–594. [Pou23b] L. Pournin,Shallow Sections of the Hypercube, Israel J. Math.255(2023), no. 2, 685–704. [Pou] L. Pournin,Deep sections of the hypercube, arX...

Pith/arXiv arXiv 1913
[2021]

Zvavitch,Gaussian measure of sections of dilates and translations of convex bodies, Adv

[Zva08] A. Zvavitch,Gaussian measure of sections of dilates and translations of convex bodies, Adv. Appl. Math.41(2008), no. 2, 247–254. Bolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, 6720 Szeged, Hungary Email address:fodorf@math.u-szeged.hu Departamento de Ingenier´ıa y Tecnolog ´ıa de Computadores, ´Area de Matem ´atica Aplicada, F ac...

2008

[1] [1]

Abel,The number of gridpoints on hyperplane sections of thed-dimensional cube, Proc

[Abe18] U. Abel,The number of gridpoints on hyperplane sections of thed-dimensional cube, Proc. Amer. Math. Soc.146(2018), no. 12, 52495355. [Ali08] I. Aliev,Siegel’s Lemma and Sum-Distinct Sets, Discr. Comput. Geom.39(2008), no. 3, 59–66. [Ali21] I. Aliev,On the volume of hyperplane sections of ad-cube, Acta Math. Hungar.163 (2021), 547–551. [AGBC] D. Al...

arXiv 2018

[2] [2]

K¨ onig and A

[KK11] H. K¨ onig and A. Koldobsky,Volumes of low-dimensional slabs and sections in the cube, Adv. Appl. Math.47(2011), no. 4, 894–907, DOI 10.1016/j.aam.2011.05.001. [KK13] H. K¨ onig and A. Koldobsky,On the maximal measure of sections of then-cube, Geo- metric analysis, mathematical relativity, and nonlinear partial differential equations, Contemp. Math...

work page doi:10.1016/j.aam.2011.05.001 2011

[3] [390]

P´ olya,Berechnung eines bestimmten Integrals., Math

[P´ ol13] G. P´ olya,Berechnung eines bestimmten Integrals., Math. Ann.74(1913), 204–212. [Pou23a] L. Pournin,Local extrema for hypercube sections, Journal d’Analyse Math´ ematique 152(2023), no. 2, 557–594. [Pou23b] L. Pournin,Shallow Sections of the Hypercube, Israel J. Math.255(2023), no. 2, 685–704. [Pou] L. Pournin,Deep sections of the hypercube, arX...

Pith/arXiv arXiv 1913

[4] [2021]

Zvavitch,Gaussian measure of sections of dilates and translations of convex bodies, Adv

[Zva08] A. Zvavitch,Gaussian measure of sections of dilates and translations of convex bodies, Adv. Appl. Math.41(2008), no. 2, 247–254. Bolyai Institute, University of Szeged, Aradi v´ertan´uk tere 1, 6720 Szeged, Hungary Email address:fodorf@math.u-szeged.hu Departamento de Ingenier´ıa y Tecnolog ´ıa de Computadores, ´Area de Matem ´atica Aplicada, F ac...

2008