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REVIEW 2 major objections 2 minor 3 references

An elementary proof of Sierksma's conjecture for seven points in the plane

T0 review · 2 major / 2 minor · reviewed 2026-05-10 · grok-4.3

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Pith's one-line read Any seven points in the plane admit four Tverberg partitions into three sets.

desk verdict Soberón gives a direct geometric case analysis proving that any seven points in the plane admit at least four Tverberg partitions into three sets. read the letter →

arxiv 2604.18485 v1 submitted 2026-04-20 math.CO

Pablo Soberón This is my paper · ORCID
classification math.CO
keywords TverbergpartitionsSierksmaconjecturesevenpointsplanegeometryconvexhullselementaryproof
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 gives a direct geometric argument showing that for any placement of seven points in the plane, it is always possible to find four distinct ways to split them into three subsets whose convex hulls share a common point. This settles the only non-trivial case of Sierksma's conjecture that has been confirmed. Earlier proofs used topological methods, while the new proof works entirely with intersections of lines and convex hulls. Readers interested in combinatorial geometry see this as evidence that small instances of the conjecture can be settled without heavy machinery.

What carries the argument

Exhaustive case analysis on the geometric configurations of the seven points and the intersection patterns of their connecting lines.

What would settle it

An explicit set of seven points in the plane whose number of Tverberg partitions into three sets is strictly less than four.

Watch

Extended reading notes

Core claim

Any seven points in the plane admit at least four Tverberg partitions into three sets, where each partition divides the points so that the three convex hulls have nonempty common intersection.

Load-bearing premise

All possible arrangements of seven points allow complete examination by cases on their intersection points without overlooked degeneracies.

Editorial extensions

If this is right

  • Sierksma's conjecture holds for seven points in the plane.
  • Elementary geometry suffices to count the partitions in this small case.
  • Topological tools are not required for verifying the conjecture when the number of points is seven.

Reading between the lines

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

  • Similar case-by-case geometric arguments might apply to eight or nine points in the plane.
  • The result hints that the minimal number of Tverberg partitions grows steadily with the number of points.
  • Direct verification on convex-position examples can serve as a quick consistency check.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 2 minor

Summary. The paper presents an elementary geometric proof that any seven points in the plane admit four Tverberg partitions into three sets. This establishes the only confirmed non-trivial case of Sierksma's conjecture, using direct geometric arguments rather than the topological methods employed in earlier proofs by Stephan Hell.

Significance. If the result holds, this provides a simpler and more elementary proof for this instance of the conjecture, which may help in understanding Tverberg partitions geometrically and could inspire similar approaches for other cases or related problems in discrete geometry. The avoidance of topological machinery is a notable strength for accessibility.

major comments (2)
  1. [§3 (case analysis)] The proof proceeds via classification of geometric arrangements and enumeration of candidate Tverberg partitions, but lacks an explicit perturbation argument or reduction lemma showing that the count of four cannot drop in degenerate configurations (e.g., three or more points collinear, or the common intersection point lying on an edge or vertex of a convex hull). This is load-bearing for the universal claim.
  2. [§4 (enumeration of partitions)] When multiple partitions share the same intersection point or when points are in special position, the counting argument in the enumeration must remain valid; the manuscript does not appear to contain a separate verification or invariance statement for these overlaps.
minor comments (2)
  1. [Abstract] The abstract is concise but could briefly note the proof technique (exhaustive geometric case analysis) to orient readers familiar with the topological proofs.
  2. [§2 (preliminaries)] Notation for Tverberg partitions (e.g., how the three sets and their common point are denoted) should be introduced once and used consistently throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough reading of our manuscript and the insightful comments. We address the major comments point by point below and outline the revisions we plan to make.

read point-by-point responses

  1. Referee: [§3 (case analysis)] The proof proceeds via classification of geometric arrangements and enumeration of candidate Tverberg partitions, but lacks an explicit perturbation argument or reduction lemma showing that the count of four cannot drop in degenerate configurations (e.g., three or more points collinear, or the common intersection point lying on an edge or vertex of a convex hull). This is load-bearing for the universal claim.

    Authors: We acknowledge that the manuscript would benefit from an explicit reduction lemma or perturbation argument to handle degenerate cases rigorously. Our case analysis in Section 3 classifies all possible configurations of seven points in the plane, including degenerate ones such as collinear points or intersection points on hull edges. In each enumerated case, we verify the existence of at least four Tverberg partitions. To address the referee's concern, we will add a lemma in the revised manuscript demonstrating that the minimum number of Tverberg partitions is preserved under small perturbations, ensuring the count does not drop in degenerate positions. This will make the argument complete for all configurations. revision: yes

  2. Referee: [§4 (enumeration of partitions)] When multiple partitions share the same intersection point or when points are in special position, the counting argument in the enumeration must remain valid; the manuscript does not appear to contain a separate verification or invariance statement for these overlaps.

    Authors: The counting in Section 4 enumerates distinct partitions of the seven points into three sets, each admitting a Tverberg point (common intersection). Even when multiple partitions share the same geometric intersection point, they are distinct as set partitions and are counted separately. The definition of a Tverberg partition is combinatorial, based on the partition of the point set, not on the uniqueness of the intersection point. We will include an additional paragraph clarifying this invariance and confirming that overlaps do not affect the lower bound of four in the revised version. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: direct elementary geometric case analysis

full rationale

The paper delivers a self-contained existence proof by exhaustive geometric classification of seven-point configurations and explicit enumeration of Tverberg partitions. No equations, parameters, or definitions reduce the claimed count of four partitions back to the input point set by construction. The sole external reference (to Hell's topological proof) is used only for contrast and is not load-bearing for the new argument. The derivation therefore stands independently of any fitted inputs, self-citations, or imported ansatzes.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The proof relies on standard convex geometry and plane topology but introduces no new free parameters or invented entities; the central claim rests on exhaustive enumeration of intersection configurations for seven points.

assumptions (2)
  • standard math Convex hulls of finite point sets in the plane are well-defined convex polygons or segments.
    Invoked implicitly when discussing Tverberg partitions and their intersections.
  • domain assumption Any finite set of points in the plane can be partitioned into three subsets whose convex hulls intersect at a point (Tverberg theorem base case).
    The paper builds on the known Tverberg theorem to count the number of such partitions.

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Cite this review

Pith. "Pith review of An elementary proof of Sierksma's conjecture for seven points in the plane." pith.science (2026). https://pith.science/paper/2604.18485

@misc{pith2026260418485,
  author       = {Pith},
  title        = {Pith review of: An elementary proof of Sierksma's conjecture for seven points in the plane},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2604.18485}},
  note         = {Machine review of arXiv:2604.18485}
}
read the original abstract

We give a new simple geometric proof that any seven points in the plane have four Tverberg partitions into three sets. This is the only confirmed non-trivial case of Sierksma's conjecture. Earlier proofs, by Stephan Hell, relied on topological arguments.

Figures

Figures reproduced from arXiv: 2604.18485 by the authors.

Figure 1
Figure 1. If x is to the left of p1p4 and to the right of p2p5 it must be in the shaded triangle. Then, the two dotted lines must leave x on the side with the most points from Y . Proof. The condition is equivalent to x ∈ C2(Y ), where Y = X \ {x}. Order the points of Y clockwise around x as p1, . . . , p6. Birch’s classic argument shows that {p1, p3, p5}, {p2, p4, p6}, {x} is a Tverberg partition. This is because any half-pl… view at source ↗
Figure 2
Figure 2. Example of the three doubly-covered sectors. It is im￾possible to separate x from a triangle with a vertex on each sector since the side that contains x would also contain a full doubly￾covered sector. Case 3. C is 2-dimensional and |C ∩ X| = 1. In this case, the point in X ∩ C defines two (3, 3, 1) Tverberg partitions. We have at least two vertices of C that are not in X, which gives us two additional (3, 2, 2) Tve… view at source ↗

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    9, 703–705

    [Avi93] David Avis,The m-core properly contains the m-divisible points in space, Pattern recognition letters14(1993), no. 9, 703–705. [Bir59] B. J. Birch,On3Npoints in a plane, Proc. Cambridge Philos. Soc.55(1959), 289–293. MR109315 [BK22] Imre B´ ar´ any and Gil Kalai,Helly-type problems, Bull. Amer. Math. Soc. (N.S.)59 (2022), no. 4, 471–502. MR4478031 ...

    work page 1993

  2. [2]

    Combin.28(2007), no

    [Hel07] Stephan Hell,On the number of Tverberg partitions in the prime power case, Eu- ropean J. Combin.28(2007), no. 1, 347–355. MR2261824 [Hel08a] Stephan Hell,On the number of Birch partitions, Discrete Comput. Geom.40 (2008), no. 4, 586–594. MR2453329 [Hel08b] Stephan Hell,Tverberg’s theorem with constraints, J. Combin. Theory Ser. A115 (2008), no. 8,...

    work page Pith review arXiv 2007

  3. [3]

    [Tve66] Helge Tverberg,A generalization of Radon’s theorem, J

    Mimeographed notes. [Tve66] Helge Tverberg,A generalization of Radon’s theorem, J. London Math. Soc.41 (1966), no. 1, 123–128. [VˇZ93] Aleksandar Vuˇ ci´ c and Rade T.ˇZivaljevi´ c,Note on a conjecture of Sierksma, Discrete Comput. Geom.9(1993), no. 4, 339–349. [Whi17] Moshe J. White,On Tverberg partitions, Israel J. Math.219(2017), no. 2, 549–553. MR3649...

    work page 1966

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Reviewed May 10, 2026 · model on record in the stance chip above.