The layer number of grids

Alexander Hsu; Bo Peng; Gergely Ambrus; Shiyu Jan

REVIEW 2 major objections 4 minor 1 cited by

The layer number of grids

T0 review · 2 major / 4 minor · reviewed 2026-08-27 · deepseek-v4-flash

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Pith's one-line read This paper proves new upper bounds for the layer number of integer grids: peeling the d-dimensional grid takes at most O(n^(d−9/11)) rounds for d≥3, with the strongest case d=3 at O(n^(24/11)).

desk verdict First non-trivial upper bounds for higher-dimensional grid peeling, but the key counting lemma is proved by a false identity and an invalid inequality; the result is very likely salvageable with a standard Möbius-inversion fix. read the letter →

arxiv 2009.13130 v1 pith:MHIP7S3N submitted 2020-09-28 math.MG math.CO

Gergely Ambrus , Alexander Hsu , Bo Peng , Shiyu Jan This is my paper · ORCID

classification math.MGmath.CO MSC 52C4568U0552A05

keywords layernumberconvexhullpeelingintegergridlayerslatticepolytopesprimitivevectorsdepth

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 asks a simple question: if you repeatedly strip the vertices of the convex hull from the integer grid $[n]^d$, how many stripping rounds are needed to delete every point? It proves that for every $d\ge1$ the answer is at least $\Omega(n^{2d/(d+1)})$, and for $d\ge3$ it is at most $O(n^{d-9/11})$. These are the first nontrivial upper bounds for the layer number of grids in dimensions three and higher. The quantitative gap between the two exponents is what the paper leaves open; the conjectured sharp answer would be $\Theta(n^{2d/(d+1)})$.

What carries the argument

Primitive normal directions with categories. A primitive vector $v$ has category $k$ when the two supporting hyperplanes orthogonal to $v$ meet the current layer polytope in $k$-dimensional faces. The proof counts, over the first $M$ peeling steps, how often each primitive direction can appear as category 0 or 1 (each such appearance consumes at least one or two hyperplanes through grid points, bounded by $dn\mu$), so each direction must spend most of the first $M$ steps in category 2. Euler's formula then ties category-2 directions to vertices: $f_0(P_i)\ge c_{2,i}$. A separate lemma isolates a large subset $V'_\mu$ of primitive vectors not perpendicular to any short lattice vector, which bounds possible edge length and improves the counting.

What would settle it

Compute $\tau([n]^3)$ exactly for increasing $n$ and compare it to $n^{24/11}$; if the ratio $\tau([n]^3)/n^{24/11}$ fails to stay bounded, the upper bound is false. Alternatively, find a cubic shell or sub-grid that, after its points become coplanar, requires more than $O(n^{d-9/11})$ additional peeling steps.

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Extended reading notes

Core claim

The central discovery is a counting identity for directions rather than points. During the first $M$ peeling steps of $[n]^3$, the authors show that the total number of vertices of the convex layer polytopes is at least the number of category-2 primitive normal directions, and then lower-bound that count by $\gamma n^2\mu^{11/3}$ for a suitable parameter $\mu$. Choosing $\mu = \Theta(n^{3/11})$ makes this vertex count exceed $n^3$, the total number of grid points, which forces the process to terminate within $O(n^{24/11})$ steps. The same argument, combined with a lemma on primitive vectors avoiding short lattice vectors and an induction on cubic shells, yields $O(n^{d-9/11})$ for all $d\ge3$.

Load-bearing premise

The upper-bound proof counts peeling steps only while the current layer polytope is non-degenerate; if after the remaining points become coplanar or collinear the peeling takes many extra steps, the counted vertex total $M$ could fall short of $n^3$ and the stated $O(n^{d-9/11})$ bound would not follow.

Editorial extensions

If this is right

For the three-dimensional grid, the peeling process terminates in $O(n^{24/11})$ steps, improving the paper's own earlier $O(n^{9/4})$ bound.
For every dimension $d\ge3$, the same type of bound $O(n^{d-9/11})$ holds, giving the first nontrivial upper bound on layer numbers of higher-dimensional grids.
Combined with the lower bound $\Omega(n^{2d/(d+1)})$, the true order of $\tau([n]^d)$ lies between these exponents for every $d\ge3$.
The lower-bound argument works in all dimensions, so the gap to the conjectured sharp value is now identified as a purely upper-bound problem.

Reading between the lines

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

If the authors' heuristic that convex layer polytopes have only short edges is correct, the category-1 counting can be tightened to $O(n\mu)$, which would yield the sharp bound $O(n^{3/2})$ for $d=3$; this is an extension, not proven in the paper.
A computational experiment on $[n]^3$ for moderate $n$ could test whether the vertex count in the first $O(n^{24/11})$ steps really dominates $n^3$, and whether the lower-dimensional tail after the remaining set becomes coplanar or collinear is short.
The planar grid peeling process has been observed numerically to approach a curve-shortening flow; the new 3D bounds raise the analogous continuum-limit question for convex hull peeling in space, though the paper does not address it.

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Formalized claims in Lean

Claim #1: The central discovery is a counting identity for directions rather than points. During the first $M$ peeling steps of $[n]^3$, the authors show that the total number of vertices of the convex layer polytopes is at least the number of category-2 primitive normal directions, and then lower-bound that count by $\gamma n^2\mu^{11/3}$ for a suitable parameter $\mu$. Choosing $\mu = \Theta(n^{3/11})$ ma

/-- @claim 1 The central discovery is a counting identity for directions rather than points. During the first $M$ peeling steps of $[n]^3$, the authors show that the total number of vertices of the convex layer polytopes is at least the number of category-2 primitive normal directions, and then lower-bound that count by $\gamma n^2\mu^{11/3}$ for a suitable parameter $\mu$. Choosing $\mu = \Theta(n^{3/11})$ ma -/
def central_claim : Prop :=

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The first thing to know: this is a real result that needs a repair, not a dud. It gives the first non-trivial upper bounds on the layer number of [n]^d for d≥3, and the lower bound via Andrews is clean. The upper-bound machinery is new: categorize primitive directions by face dimension, count how many peeling steps a direction can spend in each category, and use a short-vector-free subset of directions to reduce the edge-length cost in 3D. Theorem 4's shell induction is a sensible way to avoid McMullen's facet explosion. That part of the paper is honest and worth reading.

The soft spot is Lemma 3, and it is not cosmetic. The identity |V_m| = ∑_{i=1}^m J_{d-1}(i) is false: for d=3, m=1 the left side is 7 and the right side is 1, because V_m contains primitive vectors with zero coordinates and several choices of which coordinate realizes the maximum. The next inequality also mishandles the sign of μ(a); the step from floor bounds to a lower bound is not valid termwise. The statement |V_m|=Θ(m^d) is true, and the standard Möbius-inversion count over all nonzero vectors in [0,m]^d repairs it. So this is a proof gap in a key lemma, not a false theorem, and I would not desk-reject over it.

The degenerate-tail issue the reader flagged is secondary. Definition 2 only applies when the current layer polytope is full-dimensional, and Theorems 2 and 3 don't explicitly bound the final coplanar or collinear tail. But the tail has at most n^{d-1} points, which is swallowed by O(n^{d-9/11}) for d≥3. The authors should add one sentence.

One more small gap: Theorem 4 assumes the layer number of an arbitrary subset A∩H is bounded by the grid bound. That follows from monotonicity of layer number under inclusion—vertices of a superset that lie in the subset are vertices of the subset—but the paper doesn't state it. Easy fix.

Bottom line: the upper bounds are still far from Conjecture 1, so the paper won't reorganize the field, but for d≥3 it is the first non-trivial word. Send it to a referee; ask the referee to check the repaired Lemma 3 carefully. I would cite it once it is fixed.

Referee Report

2 major / 4 minor

Summary. The paper studies the layer number (convex peeling depth) of the d-dimensional integer grid [n]^d. It proves a general lower bound τ([n]^d) ≥ Ω(n^{2d/(d+1)}) using Andrews' bound on the number of vertices of a convex lattice polytope, and upper bounds τ([n]^d) ≤ O(n^{d−9/11}) for every d ≥ 3, with the base case d = 3 at O(n^{24/11}). The upper-bound strategy follows Har-Peled and Lidický: for a given primitive normal direction, bound the number of peeling steps in which that direction can be of category 0 or 1, then use Euler's formula and a census of primitive directions to force many vertices to be removed. A strengthened d = 3 bound uses a lemma asserting that many primitive directions are not perpendicular to short lattice vectors, which shortens the category-1 lifetime.

Significance. If the results hold, they are the first nontrivial upper bounds for τ([n]^d) in dimensions d ≥ 3 and provide the first evidence beyond the planar case for the conjecture τ([n]^d) = Θ(n^{2d/(d+1)}). The lower-bound argument is elegant and gives the same exponent as Dalal's random-point result. The paper's main conceptual contribution is the quantitative use of primitive normal directions and the short-vector lemma. The proofs are conventional, with no fitted parameters and no circular dependencies. The principal weaknesses are that Lemma 3, which supplies the Θ(μ^d) census of primitive vectors, is proved via an invalid identity and an invalid inequality, and that the category counting in Theorems 2 and 3 does not explicitly handle lower-dimensional layer polytopes. Both issues appear repairable without changing the statements.

major comments (2)

[§2.2, Lemma 3]
[Definition 2; proofs of Theorems 2 and 3]

minor comments (4)

[§2.1, Theorem 1]
[§2.3, Theorem 4]
[§2.2, proof of Theorem 2]
[Throughout]

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the layer-number bounds are derived from external lattice-geometric and polytope theorems, with no fitted parameter renamed as a prediction.

full rationale

The paper's derivation chain is self-contained with respect to circularity. The lower bound (Theorem 1) follows from Andrews's external vertex bound f0(P)=O(Vol(P)^{(d-1)/(d+1)}) applied to the layer polytopes, whose volumes are at most n^d. The upper-bound arguments use the Har-Peled--Lidicky planar peeling method [9] only as a template; the quantitative estimates are built from Lemma 1 (bounding primitive normals by determinants), Lemma 2 (at most dnμ hyperplanes per normal direction), Lemma 3 (|V_μ|=Θ(μ^d), obtained from Jordan's totient function and Möbius inversion cited to Andrica--Piticari and Hardy--Wright), and Lemma 4 (a short-vector deletion argument). Theorems 2 and 3 combine these lemmas with Euler's polyhedron formula and category counts, and then choose the auxiliary parameter μ by balancing the final inequalities; the choice is an internal optimization variable, not a parameter fitted to the target layer number. Theorem 4 is a straightforward induction on dimension using the d=3 case. No theorem, quantity, or 'prediction' reduces by construction to the layer number τ([n]^d), and no load-bearing claim is justified solely by a self-citation chain. The invalid estimate flagged in the proof of Lemma 3 is a correctness gap in the displayed inequalities, not a circularity: the lemma itself is asserted as an external arithmetic fact and the proof error does not make the conclusion an input to itself. Accordingly, the honest finding is no significant circularity.

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

No free parameters are fitted to data; the auxiliary μ is an analysis parameter eliminated in the final bound. The proof relies on standard external results and on geometric assumptions about symmetry and non-degeneracy of layer polytopes, with the latter not fully discharged.

assumptions (7)

standard math Andrews's bound on vertices of convex lattice polytopes: f0(P) ≤ O(Vol(P)^{(d-1)/(d+1)})
Used in Theorem 1 to bound how many grid points can be removed in a single peeling step.
standard math Euler's polyhedron formula f0 - f1 + f2 = 2 and f1 ≥ (3/2) f2 for 3-dimensional polytopes
Used in Theorem 2 to derive f0(P_i) ≥ c2,i from facet counts.
domain assumption The layer polytopes Pi remain centrally symmetric throughout peeling
Stated in Section 1.2; used to pair opposite supporting hyperplanes for each normal v.
domain assumption The layer polytope P remains non-degenerate at every step where categories are counted
Definition 2 explicitly assumes non-degeneracy; Theorems 2 and 3 do not treat the lower-dimensional tail, leaving a gap.
standard math The count of primitive vectors in V_μ satisfies |V_μ| = Θ(μ^d)
Needed in Theorems 2 and 3; the statement is a standard lattice-point fact, but the proof in Lemma 3 is invalid as written.
domain assumption The restriction of the d-dimensional peeling process to a supporting face equals the (d-1)-dimensional peeling process
Used in Theorem 4's induction on boundary hyperplanes.
standard math Layer number is monotone under inclusion: if B⊂C then τ(B) ≤ τ(C)
Used implicitly in Theorem 4 to bound peeling of arbitrary subsets A∩H by the layer number of [n]^{d-1}.

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Pith. "Pith review of The layer number of grids." pith.science (2026). https://pith.science/paper/MHIP7S3N

@misc{pith2026200913130,
  author       = {Pith},
  title        = {Pith review of: The layer number of grids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MHIP7S3N}},
  note         = {Machine review of arXiv:2009.13130}
}

abstract

The peeling process is defined as follows: starting with a finite point set $X \subset \mathbb{R}^d$, we repeatedly remove the set of vertices of the convex hull of the current set of points. The number of peeling steps needed to completely delete the set $X$ is called the layer number of $X$. In this paper, we study the layer number of the $d$-dimensional integer grid $[n]^d$. We prove that for every $d \geq 1$, the layer number of $[n]^d$ is at least $\Omega\left(n^\frac{2d}{d+1}\right)$. On the other hand, we show that for every $d\geq 3$, it takes at most $O(n^{d - 9/11})$ steps to fully remove $[n]^d$. Our approach is based on an enhancement of the method used by Har-Peled and Lidick\'{y} for solving the 2-dimensional case.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Sharp Bounds For The Layer Number of Integer Grids
math.CO 2026-07 accept novelty 7.0 of 10

The convex-layer number of the d-dimensional integer grid [n]^d is Theta(n^{2d/(d+1)}) for every fixed d.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

[1]

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K. Dalal. Counting the onion. Random Structures Algorithms 24 (2004), no. 2., 155–165

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Eppstein, S

D. Eppstein, S. Har-Peled, and G. Nivasch. Grid Peeling and the Aﬃne Curve-Shortening Flow . Exp. Math. 29 (2020), no. 3, 306–316

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S. Har-Peled and B. Lidick´ y. Peeling the Grid. SIAM J. Discrete Math. 27 (2013), no. 2.,

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G. Hardy and E. Wright. The Theory of Numbers. Oxford University Press, London, England, 4th edition, 1965

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Pith tools

Reviewed August 27, 2026 · model on record in the stance chip above.

[1] [1]

Alvarez, K

V. Alvarez, K. Bringmann, R. Curticapean, and S. Ray. Counting triangulations and other crossing-free struc- tures via onion layers. Discrete Comput. Geom. 53 (2015), no. 4, 675–690. 6

work page 2015

[2] [2]

G.E. Andrews. A lower bound for the strictly convex body with many boundary points. Trans. Amer. Math. Soc. 106(1993), no. 2, 270–279

work page 1993

[3] [3]

Andrica and M

D. Andrica and M. Piticari. On Some Extensions of Jordan ’s Arithmetic Functions. ICTAMI (2003)

work page 2003

[4] [4]

B´ ar´ any.Random points and lattice points in convex bodies

I. B´ ar´ any.Random points and lattice points in convex bodies. Bull. Amer. Math. Soc. 45 (2008), 339–365

work page 2008

[5] [5]

Chazelle

B. Chazelle. On the convex layers of a planar set. IEEE Trans. Inform. Theory 31 (1985), no. 4, 509–517

work page 1985

[6] [6]

W.F. Eddy. Convex Hull Peeling. In: Caussinus H., Ettinger P., Tomassone R. (eds), COMPSTAT 1982, 5th Symposium held at Toulouse 1982. Physica, Heidelberg, 1982

work page 1982

[7] [7]

K. Dalal. Counting the onion. Random Structures Algorithms 24 (2004), no. 2., 155–165

work page 2004

[8] [8]

Eppstein, S

D. Eppstein, S. Har-Peled, and G. Nivasch. Grid Peeling and the Aﬃne Curve-Shortening Flow . Exp. Math. 29 (2020), no. 3, 306–316

work page 2020