Pith. sign in

REVIEW 2 major objections 4 minor 34 references

On the volume of non-central sections of a cube

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every off-center hyperplane slice of the unit cube has volume at least 1/17, in every dimension.

desk verdict Good paper, true-looking results, but a typo in Proposition 5.4 breaks the printed proof of Theorem 1.2; it is fixable and worth refereeing. read the letter →

arxiv 1908.09358 v2 pith:NUHKNX7L submitted 2019-08-25 math.MG math.PR

Hermann König , Mark Rudelson This is my paper · ORCID
classification math.MGmath.PR MSC 52A3852A4052A20
keywords non-centralsectionsunitcubehyperplanedimension-freelowerboundspolydiscslicingproblemOrliczspacedualitytailestimates
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

This paper proves that off-center slices of the unit cube cannot become arbitrarily thin. For every unit vector $a\in\mathbb{R}^n$, the hyperplane $\{x:\langle x,a\rangle=1/2\}$ cuts the cube $Q_n=[-1/2,1/2]^n$ in a section of volume $A_{\mathbb R}(a,1)>1/17$, independent of the dimension $n$ and of the direction $a$; for complex polydiscs the analogous constant is $1/27$. More generally, for every codimension $d$ there is a positive $\varepsilon(d)$ such that any affine section of codimension $d$ at distance at most $1/2$ from the origin has volume at least $\varepsilon(d)$. The distance $1/2$ is maximal, since beyond it one can choose parallel sections that miss the cube entirely. These bounds turn the cube into one of the few bodies for which all sections in the guaranteed non-empty range are controlled from below.

What carries the argument

The load-bearing machinery is a probabilistic representation of section volumes. With independent random vectors $U_j$ uniformly distributed on $S^2$ in the real case and on $S^3$ in the complex case, the volume of the hyperplane section is $A_{\mathbb R}(a,t)=\int_{|\sum a_jU_j|\ge t} dP/|\sum a_jU_j|$ and $A_{\mathbb C}(a,t)=\int_{|\sum a_jU_j|\ge t} dP/|\sum a_jU_j|^2$. The identity $|\sum a_jU_j|^2=1+2S$, with $S=\sum_{i<j}a_ia_j\langle U_i,U_j\rangle$, turns the threshold event into $S\ge0$; the paper controls $P(S\ge0)$ from below via a Laplace-transform estimate and duality in Orlicz spaces, and controls the upper tail of $|\sum a_jU_j|$ via a Khintchine-type inequality for rotationally invariant vectors. For general codimension $d$, the section volume is interpreted as the density of a projected uniform random point, and the classical lower bound for central sections together with the known upper bound for central sections complete the reduction.

What would settle it

Check Proposition 5.4 numerically for $k=3$, $t=1.1$: the printed bound is $1.1^3\exp(3/2-3/(2\cdot1.1^2))\approx1.73$, which is greater than 1 and therefore cannot be a probability tail; the integration in Section 6 needs a genuine exponential tail near $t=1$, so replacing the printed bound by any valid tail $\le C_k e^{-c_k t^2}$ would either restore the proof with adjusted constants or reveal a genuine obstruction.

Watch

Extended reading notes

Core claim

The central claim is that dimension does not enter any lower bound for non-central sections of the cube. Specifically, Theorem 1.2 asserts $A_{\mathbb R}(a,1)>1/17$ for every unit $a\in\mathbb{R}^n$ and $A_{\mathbb C}(a,1)>1/27$ for the polydisc in $\mathbb{C}^n$, while Theorem 1.1 asserts that for each $d$, all $(n-d)$-dimensional affine sections at distance at most $1/2$ have volume $\ge\varepsilon(d)>0$. The proof derives integral representations for these volumes over products of unit spheres and bounds the resulting probabilities from below and above. The constants are not claimed to be optimal: Remark 6.1 shows they cannot be improved by more than factors $\simeq5.1$ (real) and $\simeq7.1$ (complex), since the diagonal-direction limits are $\sqrt6/(\pi e^3)\simeq0.3084$ and $2/e^2\simeq0.2707$.

Load-bearing premise

The numerical content of Theorem 1.2 depends on the upper-tail estimate in Proposition 5.4; as printed, that estimate gives a bound larger than 1 for all $t>1$, so it cannot support the integration that produces $1/17$ and $1/27$.

Editorial extensions

If this is right

  • For every dimension $n$ and every unit direction $a$, the slice $Q_n\cap\{x:\langle x,a\rangle=1/2\}$ has $(n-1)$-volume strictly larger than $1/17$, so high-dimensional cubes have uniformly thick off-center hyperplane sections.
  • Because the bound is independent of $n$, the result controls all non-central hyperplane sections simultaneously as $n$ grows, complementing the known upper bounds by the same dimension-free character.
  • The distance $1/2$ is the entire non-empty range: for any subspace $E$ and any $v\in E^\perp$ with $|v|\le1/2$, the parallel section has volume at least $\varepsilon(d)$, while for $|v|>1/2$ some choices give empty sections.
  • For every codimension $d$ a constant $\varepsilon(d)$ exists, so the phenomenon is not special to hyperplanes.
  • In the complex case, the same argument gives $A_{\mathbb C}(a,1)>1/27$ for polydisc hyperplane sections, a dimension-free lower bound in $\mathbb{C}^n$.

Reading between the lines

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

  • The sphere-integral method should extend to other rotationally symmetric bodies whose section volumes admit Bessel-function or Fourier representations, giving dimension-free lower bounds for non-central sections of such bodies in their guaranteed non-empty distance range.
  • The true infimum of $A_{\mathbb R}(a,1)$ is probably larger than $1/17$; the diagonal-direction limit $\sqrt6/(\pi e^3)$ and the exact small-dimension values suggest that the extremal direction may vary with $n$.
  • Since the printed upper-tail estimate of Proposition 5.4 is vacuous for $t$ close to $1$, a necessary repair is a valid exponential tail bound with constants depending only on $k$; if such a bound holds, the qualitative theorem survives with possibly different constants.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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 lower bounds for the volume of non-central sections of the unit-volume cube Q_n and of the complex polydisc. The first main result, Theorem 1.1, asserts that for every codimension d there is a constant ε(d) > 0 such that every affine (n−d)-plane at distance at most 1/2 from the origin cuts a section of volume at least ε(d), uniformly in the ambient dimension n and the direction of the section. The second main result, Theorem 1.2, gives explicit uniform bounds for hyperplane sections at distance exactly 1/2: A_R(a,1) > 1/17 in the real case and A_C(a,1) > 1/27 in the complex case, for every unit direction a. The proofs combine a probabilistic representation of section volumes as integrals over independent spherical vectors, exponential tail estimates obtained via Orlicz-space duality, and known results of Vaaler, Ball, and Veraar. The paper also contains a new method for bounding probabilities of the form P(|Σ a_j U_j| ≥ 1) for spherical vectors U_j, which may be of independent interest.

Significance. If the central claims hold, the paper provides the first explicit dimension-free lower bounds for non-central hyperplane sections of the cube and polydisc at the maximal distance where sections can be nonempty, complementing the classical central-section results of Vaaler and Ball. The constants are derived rather than numerically fitted, and the proof introduces a technically interesting Orlicz-space duality argument for spherical sums that goes beyond the standard Hanson-Wright framework. The lower bounds are not optimal, as the discussion in Remark 6.1 shows, but the qualitative uniformity in dimension and direction is the main contribution.

major comments (2)
  1. [§5, Proposition 5.4; §6, real and complex cases] The tail bound in Proposition 5.4 is printed with the wrong exponent: the statement P(|Σ a_j U_j| ≥ t) ≤ t^k exp(k/2 − k/(2t^2)) is vacuous for every t > 1, since the exponent k/2 − k/(2t^2) is positive and the right-hand side exceeds 1. The displayed minimization leading to the bound actually yields t^k exp(k/2 − k t^2/2); the factor t^{-2} in the exponent is a typographical error. This is not a cosmetic issue because Section 6 relies on the bound as a genuine tail estimate: the integral ∫_{t0}^∞ v exp(3/2 − 3/(2v^2)) dv, used in the real case, diverges, and the claimed equality with (1/3)exp(3/2 − 3/(2t0^2)) is false. The numerical constants 0.06011 and 0.03789, and the equations defining t0 and t1, correspond to the corrected exponent −(k/2)t^2, not to the printed inequality. The argument can likely be repaired by a local correction, but as written the proof of Theorem 1.2 does not go through.
  2. [§6, real and complex computations] The derivation of the explicit lower bounds uses a relationship between the threshold t0 (or t1) and the probability p0 (or p1) that is stated with the incorrect tail expression. The displayed equations t0^3 exp(3/2 − 3/(2t0^2)) = p0 and t1^4 exp(2 − 2t1^2) = p1 are inconsistent with the subsequent integration: the numerical values t0 ≈ 1.9182 and t1 ≈ 1.7657 solve the equations with the exponent −(3/2)t^2 and −2t^2, respectively. The printed versions make the constants p0 and p1 incompatible with the stated t0 and t1, so the numerical verification of the inequalities must be redone with the corrected tail bound.
minor comments (4)
  1. [§1, Introduction] The phrase 'probabiliy' in the reference [He] (Hensley, 'Slicing the cube in Rn and probabiliy') appears to be a misspelling of 'probability'; the reference should be checked against the original publication.
  2. [§4, Lemma 4.2] In the proof of Lemma 4.2, the inequality ∏_{j=2}^n (1+y_j) ≥ 1 + ∑_{j=2}^n y_j is stated for y_j ∈ (−1,1) 'having the same sign'; in the application all y_j are nonnegative, so the condition is satisfied, but this should be made explicit to avoid ambiguity.
  3. [§5, proof of Proposition 5.1, part (b)] The counting argument for the fourth moment E(S^4) is compressed: the sentence about '6 times' and the bound 24(∑ a_i^2 a_j^2 a_l^2 a_m^2) ≤ 4(∑ a_i^2 a_j^2)^2 is hard to follow. A short derivation of the displayed inequality would improve readability.
  4. [§6, real case] After correcting the tail exponent, the displayed intermediate expression 'p0/t0 (1 − 1/(3t0^3))' should read 'p0/t0 (1 − 1/(3t0^2))' in the real case; the printed form appears to be a typographical error.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lower bounds are derived from independent external theorems and the self-citations are not load-bearing.

full rationale

The proofs of Theorems 1.1 and 1.2 do not reduce by construction to their inputs. The volume formulas in Proposition 3.1 are cited to Ball, Pólya, and Oleszkiewicz–Pelczynski, and Proposition 3.2 derives the spherical integral representations from those standard identities rather than assuming the desired lower bound. Proposition 5.1 uses Veraar's independent centered-variable inequality together with an explicit moment computation, and Proposition 5.4 invokes the König–Kwapien Khintchine inequality, an external published result whose stated assumptions do not include the cube-section conclusion. The upper bound cited from [KK3] is used only for context, not as an input to the lower-bound argument. The printed Proposition 5.4 tail estimate is indeed vacuous for t>1, and the Section 6 integration as printed would diverge; however, this is a correctness or typographical defect in an inequality whose intended exponent is derived, not a case of the target claim being assumed through a self-citation or a fitted parameter. No step in the derivation chain is equivalent to its own inputs by definition.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new free constants fitted to data and no new postulated entities. The proof rests on established theorems in convex geometry and probability, plus a new Orlicz-space argument for quadratic forms. The main concern is the internal typo in Proposition 5.4, not any hidden postulate.

assumptions (5)
  • standard math Every central hyperplane section of the unit cube has (n-1)-volume at least 1 (Vaaler's theorem).
    Used in Lemma 2.1 and in the compressible case of Theorem 1.1 to lower-bound central sections of lower-dimensional cubes.
  • standard math Every k-codimensional central section of the unit cube has volume at most 2^(k/2) (Ball's theorem).
    Used in Lemma 2.1 and Lemma 2.2 to upper-bound central section volumes.
  • standard math For centered real-valued X, P(X >= 0) >= (2*sqrt(3)-3) E[X^2]^2 / E[X^4] (Veraar's inequality).
    Used in Proposition 5.1(a) to prove the general-k lower bound for P(S >= 0).
  • standard math Khintchine inequality for independent uniform vectors on S^(k-1): ||sum a_j U_j||_p <= b_{p,k} |a| with best constants.
    Used in Proposition 5.4 to bound the Laplace transform of |sum a_j U_j|^2.
  • standard math The density of the projection of uniform measure on a convex body is log-concave (Brunn-Minkowski).
    Used in Lemma 2.2 and Lemma 2.4 to exploit log-concavity of the section volume function.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On the volume of non-central sections of a cube." pith.science (2026). https://pith.science/paper/NUHKNX7L

@misc{pith2026190809358,
  author       = {Pith},
  title        = {Pith review of: On the volume of non-central sections of a cube},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NUHKNX7L}},
  note         = {Machine review of arXiv:1908.09358}
}
abstract

Let $Q_n$ be the cube of side length one centered at the origin in $\mathbb{R}^n$, and let $F$ be an affine $(n-d)$-dimensional subspace of $\mathbb{R}^n$ having distance to the origin less than or equal to $\frac 1 2$, where $0<d<n$. We show that the $(n-d)$-dimensional volume of the section $Q_n \cap F$ is bounded below by a value $c(d)$ depending only on the codimension $d$ but not on the ambient dimension $n$ or a particular subspace $F$. In the case of hyperplanes, $d=1$, we show that $c(1) = \frac{1}{17}$ is a possible choice. We also consider a complex analogue of this problem for a hyperplane section of the polydisc.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Ball; Cube slicing in ^n , Proc

    K. Ball; Cube slicing in ^n , Proc. Amer. Math. Soc. 97 (1986), 465-473

    work page 1986

  2. [2]

    Ball; Volumes of sections of cubes and related problems, Geometric aspects of functional analysis, Lecture Notes in Math., 1376, 251-260, Springer, Berlin, 1989

    K. Ball; Volumes of sections of cubes and related problems, Geometric aspects of functional analysis, Lecture Notes in Math., 1376, 251-260, Springer, Berlin, 1989

    work page 1989

  3. [3]

    Ball; Volume ratios and a reverse isoperimetric inequality, J

    K. Ball; Volume ratios and a reverse isoperimetric inequality, J. London Math. Soc. 44 (1991), 351-359

    work page 1991

  4. [4]

    Barthe, A

    F. Barthe, A. Koldobsky; Extremal slabs in the cube and the Laplace transform, Adv. in Math. 174 (2003), 89-114

    work page 2003

  5. [5]

    J. Bourgain; On the distribution of polynomials in high-dimensional convex sets, Geometric aspects of functional analysis, Lecture Notes in Math., 1469, 127-137, Springer, Berlin, 1991

    work page 1991

  6. [6]

    Ben-Tal, A

    A. Ben-Tal, A. Nemirovski, and C. Roos; Robust solutions of uncertain quadratic and conic-quadratic problems, SIAM Journal on Optimization, 13(2): 535-560, 2002

    work page 2002

  7. [7]

    R. B. Boppana, R. Holzman; Tomaszewski's problem on randomly signed sums: breaking the 3/8 barrier, Electron. J. Combin. 24 (2017), no. 3, Paper 3.40, 10 pp

    work page 2017

  8. [8]

    Brazitikos, A

    S. Brazitikos, A. Giannopoulos, P. Valettas, B. H. Vritsiou, Geometry of isotropic convex bodies. Mathematical Surveys and Monographs, 196. American Mathematical Society, Providence, RI, 2014

    work page 2014

Show all 34 references
  1. [9]

    D. L. Burkholder; Independent sequences with the Stein property, Ann. Math. Stat. 39 (1968), 1282-1288

    1968

  2. [10]

    Gradstein, I

    I, S. Gradstein, I. M. Ryshik; Table of Series, Products and Integrals, VEB Deutscher Verlag der Wissensch., 1957

    1957

  3. [11]

    Hadwiger; Gitterperiodische Punktmengen und Isoperimetrie, Monatsh

    H. Hadwiger; Gitterperiodische Punktmengen und Isoperimetrie, Monatsh. Math. 76 (1972), 410-418

    1972

  4. [12]

    Hensley; Slicing the cube in ^n and probabiliy, Proc

    D. Hensley; Slicing the cube in ^n and probabiliy, Proc. Amer. Math. Soc. 73 (1979), 95-100

    1979

  5. [13]

    Holzman and D

    R. Holzman and D. J. Kleitman; On the product of sign vectors and unit vectors, Combinatorica, 12(3): 303-316, 1992

    1992

  6. [14]

    D. L. Hanson, E. T. Wright; A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist. 42 (1971), 1079-1083

    1971

  7. [15]

    K\"onig, A

    H. K\"onig, A. Koldobsky; Volumes of low-dimensional slabs and sections in the cube, Adv. in Appl. Math. 47 (2011), 894-907

    2011

  8. [16]

    K\"onig, A

    H. K\"onig, A. Koldobsky; Minimal volume of slabs in the complex cube, Proc. Amer. Math. Soc. 140 (2012), 1709-1717

    2012

  9. [17]

    K\"onig, A

    H. K\"onig, A. Koldobsky; On the maximal perimeter of sections of the cube, Advances in Math. 346 (2019), 773-804

    2019

  10. [18]

    K\"onig, S

    H. K\"onig, S. Kwapie\'n; Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors, Positivity 5 (2001), 115-152

    2001

  11. [19]

    Klartag; An isomorphic version of the slicing problem, J

    B. Klartag; An isomorphic version of the slicing problem, J. Funct. Anal. 218 (2005), 372-394

    2005

  12. [20]

    Koldobsky; Fourier Analysis in Convex Geometry, Amer

    A. Koldobsky; Fourier Analysis in Convex Geometry, Amer. Math Soc., Providence, 2005

    2005

  13. [21]

    Koldobsky; An application of the Fourier transform to sections of star bodies, Israel J

    A. Koldobsky; An application of the Fourier transform to sections of star bodies, Israel J. Math 106 (1998), 157-164

    1998

  14. [22]

    Kwapie\'n; Personal communication

    S. Kwapie\'n; Personal communication

  15. [23]

    Meyer, A

    M. Meyer, A. Pajor; Sections of the unit ball of l_q^n , J. Funct. Anal. 80 (1988), 109-123

    1988

  16. [24]

    Milman, A

    V. Milman, A. Pajor; Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed n -dimensional space, Geometric aspects of functional analysis, Lecture Notes in Math. 1376, 64-104, Springer, Berlin, 1989

    1989

  17. [25]

    Moody, C

    J. Moody, C. Stone, D. Zach, A. Zvavitch; A remark on extremal non-central sections of the unit cube, Asympt. Geometr. Anal., Fields Inst. Comm. 68, 211-228, Springer, New York, 2013

    2013

  18. [26]

    Oleszkiewicz; On the Stein property of Rademacher sequences, Prob

    K. Oleszkiewicz; On the Stein property of Rademacher sequences, Prob. and Math. Stat. 16 (1996), 127-130

    1996

  19. [27]

    K. Oleszkiewicz; On p-pseudostable random variables, Rosenthal spaces and l_p^n ball slicing, Geometric aspects of functional analysis, Lecture Notes in Math., 1807, 188-210, Springer, Berlin, 2003

    2003

  20. [28]

    Oleszkiewicz, A

    K. Oleszkiewicz, A. Pelczy\'nski; Polydisc slicing in ^n , Studia Math. 142 (2000), 281-294

    2000

  21. [29]

    P\'olya; Berechnung eines bestimmten Integrals, Math

    G. P\'olya; Berechnung eines bestimmten Integrals, Math. Ann. 74 (1913) 204-212

    1913

  22. [30]

    Rudelson, R

    M. Rudelson, R. Vershynin; Hanson-Wright inequality and sub-Gaussian concentration, Electron. Commun. Probab. 18 (2013), no. 82, 9 pp

    2013

  23. [31]

    J. D. Vaaler; A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), 543-553

    1979

  24. [32]

    Veraar; A note on optimal probability lower bounds for centered random variables, Colloq

    M. Veraar; A note on optimal probability lower bounds for centered random variables, Colloq. Math 113 (2008), 231-240

    2008

  25. [33]

    Watson; A treatise on the theory of Bessel functions, second edition, Cambridge Univ

    G. Watson; A treatise on the theory of Bessel functions, second edition, Cambridge Univ. Press, 1952

    1952

  26. [34]

    Webb; Central slices of the regular simplex, Geom

    S. Webb; Central slices of the regular simplex, Geom. Dedic. 61 (1996), 19-28

    1996

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.