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Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results

T0 review · 0 major / 2 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read For a convex polygon K the probability that n uniform points form its convex hull admits an asymptotic equivalent as n tends to infinity.

desk verdict The paper derives an asymptotic equivalent for the probability that n uniform points in a fixed convex polygon are in convex position, sharpening Bárány's general-domain bounds. read the letter →

arxiv 2410.11706 v2 pith:HNGUTD7Y submitted 2024-10-15 math.PR math.CO

Ludovic Morin This is my paper
classification math.PRmath.CO
keywords convexpositionprobabilityasymptoticequivalentpolygonuniformdistributiongeometriclarge-nlimit
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 derives an explicit asymptotic equivalent for P_K(n), the probability that n points drawn independently and uniformly at random inside a convex polygon K are in convex position. This sharpens Bárány's earlier result that held for arbitrary convex bodies and extends the authors' prior work limited to regular polygons. A reader would care because the large-n decay rate governs how rare it is for all sample points to lie on the boundary of their own convex hull inside a polygonal domain.

What carries the argument

The asymptotic equivalent of P_K(n) obtained by specializing the analysis to polygonal boundaries.

What would settle it

A direct Monte-Carlo estimate of P_K(n) for successively larger n that deviates from the claimed asymptotic expression by more than the expected statistical error.

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Extended reading notes

Core claim

We give an equivalent of P_K(n) when n→∞ for a non-flat compact convex polygon K in R^2.

Load-bearing premise

The n points are drawn independently and uniformly inside the non-flat compact convex polygon K.

Editorial extensions

If this is right

  • The same asymptotic holds for every non-flat compact convex polygon, not merely regular ones.
  • The result supplies a concrete improvement over the bounds available for general convex domains.
  • The probability P_K(n) can now be compared quantitatively across different polygonal shapes K.

Reading between the lines

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

  • The constant prefactor in the asymptotic may be computable in closed form for simple polygons such as triangles or parallelograms.
  • The same specialization technique could be tested on other convex-position probabilities that are currently known only for smooth domains.
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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

0 major / 2 minor

Summary. The manuscript derives an asymptotic equivalent for P_K(n), the probability that n i.i.d. uniform random points in a fixed non-flat compact convex polygon K ⊂ R² are in convex position, as n → ∞. The result refines Bárány's theorem (valid for general convex domains) by exploiting the polygonal boundary and extends the authors' prior work on regular polygons to the general polygonal case.

Significance. If the derivation holds, the paper supplies a sharper leading-term equivalent than the general-domain case by using the finite number of sides and vertices of K. This is a natural and useful refinement in geometric probability, consistent with the standard uniform sampling model.

minor comments (2)
  1. The abstract states that an equivalent is given but does not display the explicit form of the leading term; moving a concise statement of the main asymptotic (including the dependence on the number of sides or vertices of K) into the abstract or the first paragraph of the introduction would improve readability.
  2. Notation for the polygonal boundary (e.g., the labeling of sides or the treatment of vertices) should be introduced once in §1 or §2 and used consistently thereafter to avoid any ambiguity when the limit is taken.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of the refinement over Bárány's result, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged · score 2.0 of 10

Minor self-citation present but derivation remains self-contained

full rationale

The paper derives an asymptotic equivalent for P_K(n) as n→∞ by refining Bárány's general convex-domain result to exploit the polygonal boundary of K, yielding an explicit f_K(n). The modeling assumptions (i.i.d. uniform sampling in a compact convex set of positive area) are standard and externally verifiable. A single self-citation to the author's prior work on regular polygons is noted in the abstract but is not load-bearing for the central general-polygon claim, which introduces new analysis rather than reducing to a fitted input or self-referential definition. No equations or steps reduce the claimed equivalent to its inputs by construction.

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

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted.

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Cite this review

Pith. "Pith review of Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results." pith.science (2026). https://pith.science/paper/HNGUTD7Y

@misc{pith2026241011706,
  author       = {Pith},
  title        = {Pith review of: Probability that $n$ points are in convex position in a general convex polygon: Asymptotic results},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HNGUTD7Y}},
  note         = {Machine review of arXiv:2410.11706}
}
abstract

Let $\mathbb{P}_K(n)$ be the probability that $n$ points $z_1,\ldots,z_n$ picked uniformly and independently in $K$, a non-flat compact convex polygon in $\mathbb{R}^2$, are in convex position, that is, form the vertex set of a convex polygon. In this paper, we give an equivalent of $\mathbb{P}_K(n)$ when $n\to\infty$. This improves on a famous result of B\'ar\'any (yet valid for a general convex domain $K$) and a result we initiated in the case where $K$ is a regular convex polygon.

Figures

Figures reproduced from arXiv: 2410.11706 by the authors.

Figure 1
Figure 1. An example of K P P6 There already are asymptotic results for PGpnq in the case where G is a general convex domain. One of the most important for our work is Bárány’s, who gave in [3] a logarithmic equivalent of PGpnq: Theorem 1.1. [3] For any compact convex set G with non empty interior, lim nÑ`8 n 2 pPGpnqq 1 n “ 1 4 e 2 AP˚ pGq 3 , where AP˚ pGq :“ max Sconvex set SĂG APpSq, (1) and APpSq denotes the affine perim… view at source ↗
Figure 2
Figure 2. The set of points in red are n “ 200 uniform points conditioned to be in convex position. The boundary of their convex hull is very close to a green curve being the boundary of the domain DompDq. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Two examples of convex polygons drawn in blue, and their limit shape drawn in red. On [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: If K belongs to PT κ , the boundary of DompKq is tangent to every side of K at pj for the j th side. The triangles hached with bricks patterns are the triangle Ti , i P t1, . . . , κu. In this case, the wrκs defined in (2) satisfies @j P t1, . . . , κu, wj “ fj fj ` fj…
Figure 5
Figure 5. Figure 5: The convex canonical order (d) Q pnq G is the distribution of a n-tuple zrns taken under U pnq G , conditioned to be in CGpnq. (e) Let Nκpnq “ ␣ srκs P Zą0, such that s1 `. . .`sκ “ n and sj´1 `sj ‰ 0 for all j P t1, . . . , κu ( , the set of vectors summing to n havin…
Figure 6
Figure 6. Figure 6: Two example of PCP for some K P P6 (drawn in black). The PCPpzrnsq is drawn in blue, its side-lengths crκs are represented in blue as well, and the side-distances ℓrκs are drawn in red (on the left only). On the left picture, all side-lengths crκs are nonzero whereas i…
Figure 7
Figure 7. Figure 7: In K, an example of zrns-gon, the PCPpzrnsq and its vertices br6s, and the second and third corners (the hashed areas). Here, the size-vector is sr6s “ p1, 3, 2, 0, 1, 1q. The PCP is sort of an equivalent to the ECP that we had in the regular κ-gone case. The geometric…
Figure 8
Figure 8. Figure 8: If K is in PT κ , pick C P ZK a curve (drawn in green) that is tangent to every side of K at pj for the j th side. This curve is not necessarily the frontier of DompKq. The triangles hached with dots are the triangle Ti , i P t1, . . . , κu. Set uj “ uj pCq :“ dpvj , p…
Figure 9
Figure 9. Figure 9: In cyan, an example of polygon K whose limit shape (the frontier of DompKq, in red) is not tangent to one side. The polygon delineated by the dashed black lines is the polygon KT : the frontier of DompKq is now tangent to every side of KT . We will show that DompKq “ D…
Figure 10
Figure 10. Figure 10: The polygon Kt, drawn in blue at two different times t “ 0 and t “ t ˚, and its limit shape in red. We will prove that these limit shapes are indeed the same domain. Firstly, let us prove that the map F : t ÞÑ DompKtq is continuous for the Hausdorff topology. First, n…
Figure 11
Figure 11. Figure 11: below). The polygon KT is in PT κ`1 , and satisfies DompKq “ DompKT q (by Lemma 3.3), but for all ε ą 0, there is no N P N such that for all n ě N, we have SnpK, εq “ SnpKT , εq and thus condition (12) cannot be satisfied. X KT t “ T ą t ˚ X Kt t ě T [PITH_FULL_IMAGE…

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