On the Number of Vertices in a Hyperplane Section of a Polytope

Antonio J. Torres; Gyivan Lopez-Campos; Jes\'us A. De Loera

REVIEW 3 major objections 5 minor 28 references

On the Number of Vertices in a Hyperplane Section of a Polytope

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper proves that the cyclic polytope maximizes the number of vertices a hyperplane can cut from any convex polytope: $2(n-2)$ vertices in dimension 3 and $\lfloor n/2\rfloor\lceil n/2\rceil$ in higher dimensions.

desk verdict The main theorem is true and the paper has real new content, but the proof of Theorem 1.1 has a logical gap that needs a repair; the hypercube gap theorems are plausible but partly sketched. read the letter →

arxiv 2412.12419 v3 pith:ZBVUP7OL submitted 2024-12-17 math.CO

Jesús A. De Loera , Gyivan Lopez-Campos , Antonio J. Torres This is my paper · ORCID

classification math.CO MSC 52B0552B1252B11

keywords hyperplanesectionsconvexpolytopescyclicupperboundtheoremvertexslicesequencehypercubeslicingposetgapsincounts

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 proves a sharp upper bound theorem: for any $d$-dimensional convex polytope with $n$ vertices, every hyperplane slice has at most $2(n-2)$ vertices when $d=3$, and at most $\lfloor n/2\rfloor\lceil n/2\rceil$ vertices when $d>3$; the cyclic polytope $C_d(n)$ attains both bounds. It also describes the full set of possible slice vertex counts (the vertex slice sequence) for cyclic polytopes, and proves that hypercubes in even dimensions $d>4$ always miss the value $(d/2)\binom{d}{d/2}-1$. Since slicing by a hyperplane is a basic operation in optimization, geometry, and probability, identifying the extremal polytope and the exact missing counts answers a natural analogue of the classical Upper Bound Theorem.

What carries the argument

The argument is carried by the cyclic polytope $C_d(n)$ (the convex hull of $n$ points on the moment curve) together with a perturbation lemma, Lemma 2.1, that lets one replace any slicing hyperplane by one avoiding all vertices without decreasing the number of slice vertices. Gale's Evenness Condition then gives a complete description of how a hyperplane separates the vertices of $C_d(n)$: it leaves $a$ and $b$ vertices on the two sides and $i$ on the hyperplane, and the slice has exactly $ab + i$ vertices. For enumeration and hypercube results, the slicing poset $(P_u,\le)$ encodes all vertices and non-orthogonal edges in a direction $u$, and every slice is a maximal antichain in this poset; for the hypercube this poset is the extended hypercube poset, whose width is $\lceil d/2\rceil\binom{d}{\lfloor d/2\rfloor}$.

What would settle it

For any even dimension $d>4$, find an explicit hyperplane $H$ such that $H\cap Q_d$ has exactly $(d/2)\binom{d}{d/2}-1$ vertices; the simplest target is $d=6$, where the paper's table declares 59 a gap, so a single slice with 59 vertices would refute Theorem 1.5.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.1: among all $d$-polytopes with $n$ vertices, the maximum number $\nu(P)$ of vertices in a hyperplane section is attained by the cyclic polytope, with $\nu(C_d(n)) = 2(n-2)$ for $d=3$ and $\nu(C_d(n)) = \lfloor n/2\rfloor\lceil n/2\rceil$ for $d>3$. The proof shows, via Lemma 2.1, that a maximizing slice can be assumed to avoid all vertices of $P$, and then uses the Upper Bound Theorem and Gale's Evenness Condition to count the edges crossed by such a slice. The paper also determines the complete vertex slice sequence of $C_d(n)$: in dimension 3 it is $[2(n-2)]$, and in higher dimensions it consists exactly of values $ab + i$ with $a+b = n-i$ and $0\le i\le d$. For hypercubes, Theorem 1.4 locates the first gaps of the vertex slice sequence for every $d\ge4$, and Theorem 1.5 shows that for every even $d>4$ the number $(d/2)\binom{d}{d/2}-1$ is a gap in $VSS(Q_d)$.

Load-bearing premise

The most fragile step in the proof of Theorem 1.5 is the unproved geometric assertion that translating the unique maximizing slice of an even-dimensional hypercube back until it first passes through a vertex cannot reduce the vertex count by exactly one, because each central-level vertex has more than two incident edges in each neighboring level.

Editorial extensions

If this is right

For any 3-polytope with $n$ vertices, no hyperplane slice can have more than $2(n-2)$ vertices, and cyclic polytopes realize every count from 1 to $2(n-2)$.
In dimensions $d>3$, the maximum slice size grows like $n^2/4$, so a single hyperplane cannot cut exponentially many vertices even when the polytope itself has exponentially many.
The complete description of $VSS(C_d(n))$ shows that high-dimensional cyclic polytopes have gaps exactly when the required number $i$ of vertices lying on the slicing hyperplane exceeds $d$.
For hypercubes, the first missing vertex counts are known for all $d\ge4$, and in even dimensions the last value below the maximum is always missing, so $VSS(Q_d)$ always has a gap at the top.
The poset-antichain theorem provides a finite combinatorial certificate for every slice of a given polytope and direction, enabling complete enumeration of combinatorially distinct sections.

Reading between the lines

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

Going beyond the paper: the same extremal question can be turned around, asking how small $\nu(P)$ can be among $d$-polytopes with $n$ vertices; the paper fixes only the top end of this range.
Going beyond the paper: because the slicing poset captures every slice as a maximal antichain, the same machinery could bound the number of $k$-dimensional faces of a slice, not just vertices, moving toward the broader open problem cited in [16].
Going beyond the paper: the even-dimensional hypercube gap is parity-forced (vertex-free slices of $Q_d$ have even vertex counts), suggesting that other centrally symmetric polytopes may show analogous near-maximum gaps; a computational scan of cross-polytopes would test this.

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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

Bottom line: this is a paper worth taking seriously. The main new results—the cyclic polytope slice vertex bound, the VSS formula for cyclic polytopes, the slicing poset, and the first complete gap lists for hypercubes up to dimension seven—are real. Theorem 1.1 as stated is true, and the proof gap flagged in the stress test is genuine but repairable. The UBT does not control the number of edges cut by a single hyperplane, so the sentence "it suffices to bound ν(Cd(n))" doesn't follow. But the upper bound itself is easy: a vertex-free slice partitions the vertex set into two sides, and the slice vertices are exactly the edges crossing that cut. For d>3, that is at most floor(n/2)ceil(n/2), full stop. For d=3, Proposition 4.1 gives that the two induced subgraphs are connected, so the number of uncut edges is at least n−2; the planar graph bound f1≤3n−6 then gives crossing ≤2(n−2). So the theorem survives; the proof just needs to be rewritten. The cyclic polytope attaining the bound is also shown correctly.

The poset framework in Section 4 is a clean generalization of O'Neil's idea, and Theorem 1.2 seems correct. The hypercube gap results are the most interesting part. Theorem 1.4 and the low-dimensional table are supported by the lemmas plus code, though the code at the GitHub link is unversioned and the verification for Q6/Q7 has no certificate. That is acceptable as experimental data, but the paper should say so more explicitly. The weak spot is Theorem 1.5: Lemma 5.7 (uniqueness of the maximum antichain) is plausible and the path-counting argument is reasonable, but the final geometric step—that moving the maximum slice back to a vertex cannot drop the count by exactly one—is sketched and relies on a degree observation at central levels. That could use a full proof. Lemma 5.6 also hides a lot in case analysis and figures; a skeptical referee should check the figures.

The 3-polytope gap construction (Theorem 1.3) is modest but solid, building on stacked polytopes and connectivity.

No circularity concerns; the derivations use standard external results. Citation pattern is fine.

Who this is for: people working on polytope sections, hypercube cuts, and slice combinatorics. It deserves a serious referee. I would send it to review with the expectation of a major revision: fix the Theorem 1.1 proof, expand Theorem 1.5, and clean up the computational claims.

Referee Report

3 major / 5 minor

Summary. The paper studies the function cv_P(H), the number of vertices in a hyperplane section of a convex polytope. Its central claim (Theorem 1.1) is that among d-polytopes with n vertices the cyclic polytope maximizes ν(P), with value 2(n−2) for d=3 and ⌊n/2⌋⌈n/2⌉ for d>3. The paper also introduces a slicing poset (Theorem 1.2), constructs infinitely many 3-polytopes with gaps in their vertex slice sequence (Theorem 1.3), and analyzes the vertex slice sequence of hypercubes up to dimension 7, including two general gap theorems (Theorems 1.4 and 1.5). The methods combine Gale evenness, the Upper Bound Theorem, graph connectivity, poset width, and computational experiments with provided code.

Significance. If the main theorem is correct, it gives a clean upper bound theorem for the number of vertices in hyperplane sections of polytopes, directly addressing part of Open Problem 7.7 in [16]. The explicit VSS description for cyclic polytopes, the poset framework for enumerating slices, and the new hypercube gap data are concrete and useful contributions. The paper is constructive, offers code for the hypercube computations, and does not rely on fitted parameters or circular reasoning; previous algorithmic work is used only as a comparison. However, two load-bearing proofs are incomplete as written: the reduction in Theorem 1.1 is a non sequitur, and the proof of Theorem 1.5 contains incorrect degree counts. The main claims may well be true, but the current version does not establish them.

major comments (3)

[§2, Theorem 2.4 / Proof of Theorem 1.1] The step to bound ν(P) is not justified. The proof says that because the Upper Bound Theorem makes the cyclic polytope maximize the number of edges, it suffices to bound ν(Cd(n)). This is a non sequitur: UBT controls the total number of edges f1(P) ≤ f1(Cd(n)), whereas a slice counts only the edges crossed by one hyperplane. For d>3, f1(Cd(n)) = C(n,2), which is much larger than the claimed bound ⌊n/2⌋⌈n/2⌉, and for d=3, f1(Cd(n)) = 3n−6 is larger than 2(n−2). An additional argument is needed: for d>3, a vertex-free slice is a cut δ(S) in the 1-skeleton, so it has size at most |S|(n−|S|) ≤ ⌊n/2⌋⌈n/2⌉; for d=3 a separate planar dual or cycle bound is needed to obtain 2(n−2). Without such an argument, the upper-bound half of Theorem 1.1 is unsupported.
[§5.1, Lemma 5.7 and proof of Theorem 1.5] The characterization of maximum antichains in Lemma 5.7 is stated and proved with incorrect level indices. With Ei = {e=(v1,v2) : ||v1||+1=||v2||=i}, the path count through an edge at level i is (d−i)!(i−1)!, whose minimum for even d occurs at i = d/2 and i = d/2+1; the lemma's E⌈d/2⌉ ∪ E⌊d/2⌋ collapses to E_{d/2} for even d, and the proof's 'i∈{d/2,d/2−1}' is inconsistent with that minimum. More seriously, the degree-count observation used in the proof of Theorem 1.5 is false as written: for e={u,v}∈Ei with u at level i−1, u is incident to d−i+1 edges in Ei and to i−1 edges in E_{i−1}, not to d−i and i. Consequently the assertion that endpoints of central-level edges have more than two incident edges in each of the consecutive levels is not true as stated (for d=6, a level-2 endpoint of an E3 edge has two edges in E2). Since this is exactly the step meant to exclude a drop of exactly one vertex when translating back to a vertex-containing slice, the proof of Theorem 1.5 is incomplete.
[§5.2, Corollary 1.5.1] The exactness of Table 1 for Q6 and Q7 is certified only by the statement that all other values are realized by slices computed with the GitHub code. The paper does not specify the finite search space: which directions were used, how the authors know that these directions produce all possible values of cv_Qd, or how the enumeration was certified. Since Corollary 1.5.1 is a claim of exact VSS, the computational proof should include this information, for example a list of representative directions or an argument that the chosen set realizes every attainable slice type.

minor comments (5)

[§4, Theorem 1.2] The statement should specify that an edge is counted only when H meets its relative interior. If edges are counted by mere intersection, then a hyperplane through a vertex makes all incident edges 'intersected' at that vertex, and the resulting set is not an antichain. The proof itself uses the relative-interior convention, so the statement should be aligned with it.
[§5.1, Lemma 5.7] The set of central levels for even d should be E_{d/2} ∪ E_{d/2+1} in the statement, and the proof's minimum calculation should refer to the same two levels; the current E⌈d/2⌉ ∪ E⌊d/2⌋ is a typo that obscures the argument.
[§5.1, proof of Lemma 5.5] In the case |H−∩V(Qd)| = 3, 'N(v1)∩N(v1) = v4' should read 'N(v1)∩N(v3) = v4'.
[§2, Theorem 2.3] For d>3, the proof asserts that every triple (a,b,i) with a+b=n−i and i≤d is realizable by a suitable hyperplane, but it does not give the construction. A brief Gale-evenness argument showing how to place i points on H and split the remaining a and b vertices would make the VSS description of the cyclic polytope fully rigorous.
[§5.2, Corollary 1.5.1] Please include the version or commit of the GitHub repository and a short description of the computational enumeration so that the claim that all listed values are attained can be reproduced independently.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found: the main bounds are derived from external results (Upper Bound Theorem, Gale evenness, Steinitz, O'Neil width) and computational enumeration; the self-citation [6] is not load-bearing. The skeptical concern about Theorem 1.1's UBT step is a correctness gap, not a circularity.

full rationale

The paper's central claim, Theorem 1.1, is intended as an upper bound theorem for the number of vertices in hyperplane slices. The proof attempts to pass from the Upper Bound Theorem (external, McMullen) to a bound on slice vertices via Lemma 2.1. Even if that step is logically insufficient, it is not circular: the paper does not define the maximum slice vertex count in terms of the cyclic polytope, and it does not feed the target inequality back into its assumptions. The cyclic polytope values are computed directly from Gale's Evenness Condition, an independent classical result. The hypercube results rely on O'Neil's width result, also external, plus their own poset framework; the poset framework is not an ansatz imported from prior work but is introduced and proved in the paper. The paper cites its own prior work [6] for algorithmic context and for enumerating directions, but not to fix any constant or to prove any theorem used in the main derivation. The fragile step highlighted in the proof of Theorem 1.5 (the assertion that moving the hyperplane back cannot drop the vertex count by exactly one because central-level vertices have more than two incident edges in adjacent levels) is a geometric lemma stated without full detail; however, it does not reduce the conclusion to an input or to a fitted parameter. Likewise, the skeptical objection to the UBT step in Theorem 1.1 is a proof gap about why bounding slices of the cyclic polytope bounds all polytopes, not a circularity by definition. Overall, the derivation chain is not self-referential in a way that would make the predictions equivalent to their inputs.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

No free parameters or invented entities: the paper is purely combinatorial and uses standard external theorems plus one tooling assumption about its computational enumeration. The main derivations are grounded in external benchmarks rather than in the paper's own conclusions.

assumptions (8)

standard math Upper Bound Theorem: Cd(n) maximizes each entry of the f-vector among d-polytopes with n vertices
Used in the proof of Theorem 1.1 to transfer extremality to the cyclic polytope.
standard math Gale's Evenness Condition characterizes facets of cyclic polytopes
Used in Corollary 2.2.1 and Theorem 2.3 to determine which moment-curve points fall on each side of a hyperplane.
standard math Balinski's theorem: the graph of a d-polytope is d-connected
Used in Lemma 3.3 to lower bound slice vertex counts when a hyperplane cuts the interior of P.
standard math Menger's theorem: k-connected graphs contain k disjoint paths between any two vertices
Used in Proposition 3.1 and Lemma 5.6 for lower bounds on slice vertex counts.
standard math Steinitz's theorem: every 3-connected planar graph is realizable as a 3-polytope
Used to turn the infinite family of 5-connected planar graphs from Barnette into 3-polytopes with a gap.
standard math O'Neil's 1971 width result: the maximum number of edges cut by a slice of Qd is ceil(d/2) times C(d, floor(d/2))
Used in Section 5 to identify nu(Qd) and to anchor the uniqueness argument in Lemma 5.7.
standard math Any circuit and any cutset of a graph share an even number of edges
Used in Proposition 5.2 to prove slices avoiding vertices have even vertex count for even-dimensional hypercubes.
domain assumption The linked Python program correctly enumerates all slice vertex counts for Q6 and Q7
The completeness of Corollary 1.5.1's Q6 and Q7 rows depends on this unversioned code.

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Cite this review

Pith. "Pith review of On the Number of Vertices in a Hyperplane Section of a Polytope." pith.science (2026). https://pith.science/paper/ZBVUP7OL

@misc{pith2026241212419,
  author       = {Pith},
  title        = {Pith review of: On the Number of Vertices in a Hyperplane Section of a Polytope},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZBVUP7OL}},
  note         = {Machine review of arXiv:2412.12419}
}

abstract

We study the slices or sections of a convex polytope by affine hyperplanes. We present results on two key problems: First, we provide tight bounds on the maximum number of vertices attainable by a hyperplane slice of $d$-polytope (a sort of upper bound theorem) and discuss a new algorithm to find all sections. Second, we investigate the sequence of numbers of vertices produced by the different slices over all possible hyperplanes and analyze the gaps that arise in that sequence. We study these sequences for three-dimensional polytopes and for hypercubes. Our results were obtained with the help of large computational experiments, and we report on new data generated for hypercubes.

Figures

Figures reproduced from arXiv: 2412.12419 by the authors.

**Figure 1.** Visualization of the 1-skeletons of C3(7) and C4(7), alongside two hyperplanes (shown as purple lines) positioned to maximize the number of edge intersections in each case. Proof of Theorem 1.1. By the Upper Bound Theorem, the cyclic polytope Cd(n) maximizes the number of faces of each dimension, in particular, the number of edges among all d-polytopes with n vertices. According to Lemma 2.1, the value of ν(P) is ac… view at source ↗

**Figure 2.** For every combinatorially different pair of vertice [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** A regular tetrahedron T and the stacked tetrahedron σ(T). 7 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: The octahedron O with vertices {a, b, c, d, e, f} (left), the poset induced by direction u (center), and the poset induced by direction w (right). Proof of Theorem 1.2. Let P be a d-polytope and u ∈ R d . We show that for each u and any hyperplane H ∈ HP with normal ve…

**Figure 5.** Figure 5: Configurations when |H− ∩ V (Qd)| < 4 d − 1 d − 1 d − 2 v1 v2 v3 ... ... ... v4 ... d − 2 d − 1 d − 1 v2 v3 ... ... ... v4 d − 1 v1 d − 3 d − 2... ... ... d − 2 ... v1 v2 v3 v4 d − 2 d − 2 ... a) b) c) [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

**Figure 6.** Figure 6: Configurations when |H− ∩ V (Qd)| = 4 at least one pair of antipodal vertices (antipodal restricted to each of these sub-hypercubes, not antipodal in Qd) in different half-spaces. This would imply, by Menger’s Theorem (for more details see [5], that cvQd (H) ≥ Xr i=1 d…

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Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [16]

Combinatorics of sections of polyt opes and Coxeter groups in Lobachevsky spaces

Askold Khovanskii. Combinatorics of sections of polyt opes and Coxeter groups in Lobachevsky spaces. In The Coxeter legacy , pages 129–157. Amer. Math. Soc., Providence, RI, 2006

work page 2006

[2] [1]

Volumes of sections of cubes and related prob lems

Keith Ball. Volumes of sections of cubes and related prob lems. In Geometric Aspects of Functional Analysis: Israel Seminar (GAF A) 1987–88 , pages 251–260. Springer, 2006

work page 1987

[3] [2]

The minimum number of vertices of a simpl e polytope

David Barnette. The minimum number of vertices of a simpl e polytope. Israel Journal of Mathematics, 10:121–125, 1971

work page 1971

[4] [3]

A proof of the lower bound conjecture for convex polytopes

David Barnette. A proof of the lower bound conjecture for convex polytopes. Paciﬁc Journal of Mathematics , 46:349–354, 1973

work page 1973

[5] [4]

On generating planar graphs

David Barnette. On generating planar graphs. Discrete Mathematics, 7(3-4):199–208, 1974

work page 1974

[6] [5]

Graph theory

John Adrian Bondy and Uppaluri Siva Ramachandra Murty. Graph theory. Springer Publishing Company, Incorporated, 2008. 20

work page 2008

[7] [6]

The best ways to slice a polytope

Marie-Charlotte Brandenburg, Jes´ us De Loera, and Chia ra Meroni. The best ways to slice a polytope. Mathematics of Computation , 94(352):1003–1042, 2025

work page 2025

[8] [7]

Same average in every di rection

Imre B´ ar´ any and Gabor Domokos. Same average in every di rection. Bulletin Math´ ematique de la Soci´ et´ e des Sciences Math´ ematiques de Roumanie, 67(2):125–138, 2024

work page 2024