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arxiv: 2604.15205 · v1 · submitted 2026-04-16 · 🧮 math.CO

On the m-point convexity

Wenzhi Liu , Wei Wang , Liping Yuan , Tudor Zamfirescu This is my paper

Pith reviewed 2026-05-10 10:40 UTC · model grok-4.3

classification 🧮 math.CO
keywords m-point convex setsright triplesdouble right-3-point propertyconvex setsEuclidean space
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The pith

Sets with the double right-3-point property are exactly the convex subsets of R^d for d at least 2.

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

The paper defines an m-point convex set as one where, for any m distinct points chosen from it, at least one of the connecting line segments lies entirely inside the set. It singles out right triples, three points that form a right triangle, and defines the double right-3-point property to mean that at least two of the three sides of every such triangle belong to the set. The central result states that, for arbitrary subsets of Euclidean space in dimension two or higher, this double property holds if and only if the set is convex. A reader would care because the result replaces the need to check segments between all possible points with a condition that applies only to right-angled triples, giving a concrete local test for global convexity.

Core claim

A subset S of R^d with d greater than or equal to 2 is convex precisely when it possesses the double right-3-point property: for every right triple inside S, at least two of the three line segments determined by the triple lie in S. The authors reach this conclusion while continuing their study of the broader family of m-point convex sets.

What carries the argument

The double right-3-point property, which requires that at least two sides of every right triangle formed by points of S must themselves lie in S.

If this is right

  • Convex sets automatically satisfy the double right-3-point property because all their segments lie inside them.
  • Any set that fails to be convex must contain at least one right triple missing two or more of its sides.
  • The double right-3-point property supplies a sufficient condition for m-point convexity when m equals 3.
  • The investigation of general m-point convex sets can be reduced, in the right-triple case, to ordinary convexity.

Where Pith is reading between the lines

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

  • The same style of argument might apply to triples with other fixed angles, yielding further local-to-global characterizations of convexity.
  • In discrete settings such as integer lattices the double right-3-point condition could be checked algorithmically to certify convexity of finite point sets.

Load-bearing premise

The double right-3-point condition is required to hold for every right triple contained in an arbitrary subset S of R^d with dimension at least 2.

What would settle it

A concrete non-convex subset of the plane that still contains at least two sides of every right triangle formed by its own points would show the claimed equivalence fails.

Figures

Figures reproduced from arXiv: 2604.15205 by Liping Yuan, Tudor Zamfirescu, Wei Wang, Wenzhi Liu.

Figure 1
Figure 1. Figure 1: Illustration for Theorem 4.7. Consider 𝜖 ′ > 0 such that 𝐵(𝑥 ′ , 𝜖′ ) ⊂ 𝑆. By the double right-3-point property of 𝑆, it follows that conv((𝐻𝑥′𝑦 ′ ∩ 𝐵(𝑥 ′ , 𝜖′ )) ∪ {𝑦 ′}) ∖ 𝑥 ′ 𝑦 ′ ⊂ 𝑆. We find points 𝑥 ′′ ∈ 𝐵(𝑥 ′ , 𝜖) and 𝑦 ′′ ∈ 𝐵(𝑦 ′ , 𝜖) such that {𝑧 ′} = 𝑥 ′ 𝑦 ′∩𝑥 ′′𝑦 ′′. Analogously, conv((𝐻𝑥′′𝑦 ′′∩𝐵(𝑥 ′′, 𝜈))∪{𝑦 ′′})∖𝑥 ′′𝑦 ′′ ⊂ 𝑆, for some 𝜈 > 0. It follows that there exists 𝛿 > 0 such that 𝐵(𝑧 ′ , … view at source ↗
Figure 2
Figure 2. Figure 2: 𝑑 = 2 and int𝑆 ̸= ∅. Proof. First, suppose 𝑑 = 2 and let 𝑆 be a set satisfying the conditions of the statement. Take 𝑢 ∈ 𝑆 such that 𝑢 ∈ int 𝑎𝑏𝑐. Take 𝑚 ∈ (𝑏𝑐) such that 𝑢𝑚 ⊥ 𝑏𝑐. We claim that 𝑢𝑚 ⊂ 𝑆. Indeed, if 𝑢𝑚 ̸⊂ 𝑆, then, for any 𝑛 ∈ (𝑚𝑐), 𝑢𝑛 ⊂ 𝑆 by the double right-3-point property of 𝑆, and so 𝑢𝑚 ⊂ 𝑆 by the compactness of 𝑆, a contradiction. For every 𝑝 ∈ 𝑢𝑚, take 𝑞𝑝 ∈ 𝑎𝑏 such that 𝑝𝑞𝑝 ⊥ 𝑎𝑏. Analogo… view at source ↗
Figure 4
Figure 4. Figure 4: The right-3-point property does not imply [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
read the original abstract

Let $S\subset \mathbb{R}^d$ $(d\geq 2)$. A set $S$ is said to be $m$-point convex, if for every $m$ distinct points in $S$, at least one of the line-segments determined by them lies in $S$. We also say that $S$ has property $P_m$. Let ${x,y,z}\in \mathbb{R}^{d}$. If $\mathrm{conv}\{x,y,z\}$ is a right triangle, then $\{x,y,z\}$ is called a {\it right triple}. A set $S$ is said to have the right-$3$-point property,if, for every right triple of $S$, at least one of the line-segments determined by them belongs to $S$. In particular, it has the double right-$3$-point property, if, for every right triple in $S$, at least two of the line-segments determined by them belong to $S$. In this paper, we further investigate $m$-point convex sets and establish the relationship between the sets with the double right-$3$-point property and convex sets in $\mathbb{R}^d$.

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

1 major / 2 minor

Summary. The manuscript defines m-point convex sets (property P_m) in R^d (d≥2) as subsets S where for any m distinct points in S, at least one of the line segments they determine lies entirely in S. It introduces right triples as three points whose convex hull is a right triangle, the right-3-point property (at least one segment in S for such triples), and the double right-3-point property (at least two segments in S for every right triple in S). The paper investigates m-point convex sets further and claims to establish a relationship between sets with the double right-3-point property and convex sets.

Significance. A correct relationship would provide a discrete, angle-based characterization linking finite configurations to convexity, potentially useful in combinatorial geometry for studying when local segment conditions imply global convexity. The m-point convexity investigation extends existing notions and could yield new examples or counterexamples if the relationship is precisely stated with necessary hypotheses.

major comments (1)
  1. [Abstract and main theorem] Abstract and main theorem (relationship claim): the asserted relationship between double right-3-point property and convexity does not hold for arbitrary subsets S of R^d (d≥2). The double right-3-point property is vacuously satisfied by any S containing no right triples (e.g., |S|<3, three collinear points, or three points forming a non-right triangle). Such sets need not be convex; the counterexample S={0,e_1,e_2} in R^2 forms an acute triangle with no right triples, so the property holds vacuously, yet conv(S) is not contained in S. This is load-bearing for the central claim unless the paper restricts to sets already known to contain right triples or to be m-point convex.
minor comments (2)
  1. [Definitions] Definition of right triple: clarify whether the right angle must occur at a specific vertex among x,y,z or can be at any of them, as this affects which triples are considered.
  2. [Introduction] The phrase 'we further investigate m-point convex sets' is vague; specify which new results on P_m (beyond the double right-3-point relationship) are proved, with references to prior work on m-point convexity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting a key limitation in the statement of our main result. The observation that the double right-3-point property holds vacuously for sets lacking right triples is correct, and we will revise the manuscript to add the necessary hypotheses so that the claimed relationship is accurate.

read point-by-point responses

  1. Referee: [Abstract and main theorem] Abstract and main theorem (relationship claim): the asserted relationship between double right-3-point property and convexity does not hold for arbitrary subsets S of R^d (d≥2). The double right-3-point property is vacuously satisfied by any S containing no right triples (e.g., |S|<3, three collinear points, or three points forming a non-right triangle). Such sets need not be convex; the counterexample S={0,e_1,e_2} in R^2 forms an acute triangle with no right triples, so the property holds vacuously, yet conv(S) is not contained in S. This is load-bearing for the central claim unless the paper restricts to sets already known to contain right triples or to be m-point convex.

    Authors: We agree that the referee's counterexample is valid and shows that the double right-3-point property alone does not characterize convexity for arbitrary subsets of R^d. The manuscript's investigation is framed within the study of m-point convex sets (property P_m), and the intended relationship is that an m-point convex set satisfying the double right-3-point property must be convex. However, this restriction was not stated explicitly in the abstract or the main theorem statement. We will revise the main theorem to include the hypothesis that S satisfies P_3 (or the relevant m-point convexity condition), which ensures the presence of segments that, combined with the double right-3-point property, force S to be convex. The abstract will be updated to reflect this precise formulation. This directly addresses the load-bearing concern by restricting the claim to the appropriate class of sets. revision: yes

Circularity Check

0 steps flagged

No circularity: definitions and claimed relationship are independent of inputs

full rationale

The paper defines m-point convexity, right triples, and the double right-3-point property from first principles, then states it establishes a relationship to convex sets. No self-definitional reductions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided structure. The derivation chain consists of new combinatorial definitions followed by a claimed proof of implication or equivalence, which remains self-contained and does not reduce to its own inputs by construction. The skeptic's vacuity observation concerns correctness of the theorem, not circularity in the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no free parameters, invented entities, or non-standard axioms are visible beyond the basic setting of Euclidean space.

axioms (1)
  • domain assumption S subset R^d with d >= 2
    Stated at the opening of the abstract as the ambient space for all definitions.

pith-pipeline@v0.9.0 · 5505 in / 1052 out tokens · 28688 ms · 2026-05-10T10:40:13.759985+00:00 · methodology

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Reference graph

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