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arxiv: 2005.08292 · v2 · submitted 2020-05-17 · 🧮 math.MG · math.FA

Central diagonal sections of the n-cube

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

Pith reviewed 2026-05-24 15:22 UTC · model grok-4.3

classification 🧮 math.MG math.FA
keywords cubehyperplane sectionscentral sectionssection volumemonotonicityLaplace methodinterval arithmetic
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The pith

The volume of central hyperplane sections of the unit n-cube orthogonal to a space diagonal is strictly increasing in n for all n ≥ 3.

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

The paper proves that these specific central sections of the unit cube grow strictly larger in volume as dimension increases, starting from n=3. It combines an integral formula for the section volume with Laplace's method to handle the large-n regime, then uses interval arithmetic and automatic differentiation to bound the threshold where the asymptotic argument begins and to verify the finite number of smaller cases directly. A reader cares because the result settles a natural monotonicity question for geometrically canonical slices and shows that higher-dimensional cubes have measurably bigger central diagonal cross-sections. The argument splits the problem into an analytic tail and a computer-assisted initial segment.

Core claim

We prove that the volume of central hyperplane sections of a unit cube in R^n orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for n≥3. Our argument uses an integral formula that goes back to Pólya for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific n0. Then, using interval arithmetic and automatic differentiation, we compute an explicit bound for n0, and check the remaining cases between 3 and n0 by direct computation.

What carries the argument

Pólya's integral formula for the volume of a central cube section, whose asymptotic growth is controlled by Laplace's method and whose finite initial segment is certified by interval arithmetic and automatic differentiation.

If this is right

  • The sequence of section volumes is strictly increasing for every integer dimension at least 3.
  • Laplace's method supplies the monotonicity once n exceeds an explicit, computable threshold n0.
  • Direct rigorous numerical evaluation settles the inequality for all dimensions from 3 up to that threshold.
  • The volumes therefore form a strictly monotone sequence without exception for n≥3.

Where Pith is reading between the lines

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

  • Similar monotonicity statements might hold for central sections orthogonal to other vectors or for non-central parallel sections.
  • The same integral-plus-Laplace strategy could be tested on sections of other symmetric convex bodies such as crosspolytopes.
  • One could ask whether the normalized volumes converge to a limit and what geometric interpretation that limit carries in high dimensions.

Load-bearing premise

The interval-arithmetic and automatic-differentiation computations correctly produce a rigorous upper bound on the threshold n0 from which the Laplace-method monotonicity argument applies, and the direct numerical checks for dimensions 3 through n0 are free of rounding or implementation errors.

What would settle it

A single pair of consecutive integers n and n+1 with n≥3 for which the computed section volume at dimension n+1 is smaller than at dimension n.

Figures

Figures reproduced from arXiv: 2005.08292 by Bernardo Gonz\'alez Merino, Ferenc Bartha, Ferenc Fodor.

Figure 1
Figure 1. Figure 1: Voln−1(C n ∩ H(u0)) for 3 ≤ n ≤ 110 plotted by Mathematica. Recently, K¨onig and Koldobsky proved that, in fact, Voln−1(C n ∩ H) ≤ p 6/π for all n ≥ 2, see [KK19, Prop. 6(a)]. We also point out the recent result of Aliev [Ali20] (see also [Ali08]) about hyperplane sections of the cube, in which he proves that (3) √ n √ n + 1 ≤ I(n + 1) I(n) which is slightly less than the monotonicity of Voln−1(C n ∩ H(u0)… view at source ↗
Figure 2
Figure 2. Figure 2: I(n + 1) − I(n) for 3 ≤ n ≤ 145 plotted by Mathematica Remark. Using the same ideas as above, one could show the concavity of I(n) for n ≥ 3 [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The upper bound of Ef (t; N, m) for various (N, m) over t ∈ [0, 1.1]. 5.2. The function 1 + t 2F(t) = sin t t . Even though there are no issues with directly computing log (f(t)) using the results above, as shown in (10), we will need a more sophisticated approach in order to be able to tackle the final square root operation in the neighbourhood of zero. To that end, we rewrite expansion (11) as f(t) = 1 +… view at source ↗
Figure 4
Figure 4. Figure 4: The upper bound of Eg(t; N, m) for various values. Note that we altered the notation somewhat compared to [5] and use Taylor coefficients instead of derivatives, this should not cause confusion. 6. Derivatives of x(t) Using the combination of results of Section 5, we may attempt to evaluate x(t) and its derivatives based on the steps detailed in (10). The expansions of −6, t, and t 2 are trivial, so is the… view at source ↗
read the original abstract

We prove that the volume of central hyperplane sections of a unit cube in $\mathbb{R}^n$ orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for $n\geq 3$. Our argument uses an integral formula that goes back to P\'olya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific $n_0$. Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for $n_0$, and check the remaining cases between $3$ and $n_0$ by direct computation.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that the volume of central hyperplane sections of the unit n-cube orthogonal to a space diagonal is a strictly monotonically increasing function of n for all n≥3. The argument first applies Laplace's method to Pólya's integral formula to obtain asymptotic monotonicity for sufficiently large n, then uses interval arithmetic combined with automatic differentiation to compute an explicit finite threshold n0, and finally verifies the strict increase by direct computation on the finite range 3≤n<n0.

Significance. If the computational verification is correct, the result completes the monotonicity picture for these diagonal sections, combining analytic asymptotics with rigorous numerics in a standard and effective way for such problems in convex geometry. The explicit determination of n0 and the direct checks constitute a concrete, falsifiable bridge between the asymptotic regime and small dimensions.

major comments (2)
  1. [section describing computation of n0] Computation of n0 via IA/AD: the explicit bound on n0 obtained by interval arithmetic and automatic differentiation is load-bearing for the entire argument, since an undetected implementation or rounding error could produce an n0 that is too small and leave an unproven gap; the manuscript must supply sufficient implementation details (library, precision, code or pseudocode) to permit independent reproduction of this bound.
  2. [section on direct verification] Direct numerical verification for 3≤n<n0: floating-point or rounding errors in these checks could miss a local violation of monotonicity; the paper should state the exact arithmetic precision, the method used to certify strict inequality, and any interval enclosures employed for these finite cases.
minor comments (2)
  1. [abstract] The abstract and introduction should state the numerical value of the computed n0 explicitly.
  2. [references] The bibliography entries for the cited works of Pólya, H, and B86 should be expanded with full publication details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for the constructive comments on reproducibility. We address each major point below and will revise the manuscript to incorporate the requested details.

read point-by-point responses

  1. Referee: [section describing computation of n0] Computation of n0 via IA/AD: the explicit bound on n0 obtained by interval arithmetic and automatic differentiation is load-bearing for the entire argument, since an undetected implementation or rounding error could produce an n0 that is too small and leave an unproven gap; the manuscript must supply sufficient implementation details (library, precision, code or pseudocode) to permit independent reproduction of this bound.

    Authors: We agree that the explicit bound on n0 is central to the argument and that implementation details are required for reproducibility. In the revised manuscript we will add the specific library (or libraries) employed for interval arithmetic and automatic differentiation, the working precision, and a clear description or pseudocode of the algorithm used to compute the bound. revision: yes

  2. Referee: [section on direct verification] Direct numerical verification for 3≤n<n0: floating-point or rounding errors in these checks could miss a local violation of monotonicity; the paper should state the exact arithmetic precision, the method used to certify strict inequality, and any interval enclosures employed for these finite cases.

    Authors: We agree that the finite-range verification must be presented with sufficient rigor to certify the strict inequalities. The revised manuscript will state the arithmetic precision used, the certification method (including any interval enclosures), and how strict monotonicity is verified for 3 ≤ n < n0. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses external integral formula, standard asymptotics, and independent numerical verification.

full rationale

The paper cites Pólya's integral formula externally, applies Laplace's method to establish asymptotic monotonicity from a finite n0 onward, computes an explicit n0 bound via interval arithmetic and automatic differentiation (external computational tools), and performs direct numerical checks for n=3 to n0. No self-citations appear load-bearing, no parameters are fitted and renamed as predictions, and no step reduces the claimed monotonicity to a definition or input by construction. The argument is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof begins from a known integral representation and applies standard analytic and numerical tools; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Pólya integral formula for volumes of central cube sections
    Cited as the starting point for both the asymptotic analysis and the numerical checks.

pith-pipeline@v0.9.0 · 5662 in / 1158 out tokens · 23779 ms · 2026-05-24T15:22:41.607494+00:00 · methodology

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Reference graph

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6 extracted references · 6 canonical work pages

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