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Longest convex chains with i.i.d. points

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Pith's one-line read For n i.i.d. points in a triangle, the longest convex chain grows as c n^{1/3}, where c is a density-weighted affine arclength maximum and near-longest chains converge to its maximizers.

desk verdict Correct-looking, long, and genuinely new proof of the variational formula for longest convex chains; the additional density assumption (H2) only affects the shape-concentration theorems, not the main growth rate. read the letter →

arxiv 2608.09105 v1 pith:SVCH3LXS submitted 2026-08-10 math.PR

Erik Bates , Arnab Sen This is my paper · ORCID
classification math.PR MSC 60K3560K3760D0582B44
keywords convexchainsi.i.d.pointspositionlast-passagepercolationvariationalformulaaffinearclengthlimitshapelargedeviations
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

This paper proves the leading-order law of the longest convex chain among $n$ i.i.d. points in a triangle, for any continuous sampling density. The claim is that $L_n/n^{1/3}$ converges to a universal constant times a maximum of a density-weighted affine arclength functional, so the density enters the problem only through a variational constant, and that all near-longest chains concentrate around the maximizing curves. If the paper is right, a natural geometric extremal statistic that was solved only for uniformly distributed points now has a complete first-order theory: the growth scale is $n^{1/3}$ (against $n^{1/2}$ for the analogous longest monotone chain problem), the limiting shape is the optimizer of the variational formula, and even the rare event that all $n$ points form one convex chain has a known rate. A sympathetic reader would care because the proof supplies the same kind of variational description, with optimizer concentration, that has been established for longest monotone chains, and it connects to the affine-perimeter functionals known from limit shapes of random convex lattice polygons.

What carries the argument

The load-bearing object is the functional $J_p(f)=\int_0^1 (f''(x)p(x,f(x)))^{1/3}\,dx$ together with the tangency triangles attached to a candidate curve. For a convex $f$ and a short interval $[x,x+\varepsilon]$, the tangency triangle is the triangle bounded by the two tangent lines of $f$ at the endpoints and the secant joining them; its area is $\frac18 f''(x)\varepsilon^3+o(\varepsilon^3)$, so the mean number of sample points inside it is about $\frac{n}{8}f''(x)p(x,f(x))\varepsilon^3$. The lower bound partitions a curve into many such triangles, builds a longest chain inside each by comparison with the uniform case, and glues them with a concatenation lemma. The upper bound shows conversely that every convex chain is captured by the tangency triangles of some piecewise-convex function $\tilde f$ drawn from a family of only $e^{o(n)}$ possibilities, so the chain length is controlled by $J(\tilde f)$ up to negligible error, and a repair step converts $\tilde f$ into a genuinely convex function with nearly the same $J$-value. Upper semicontinuity of $J$ on a compact metric space of convex functions guarantees that maximizers exist; the separation penalty (Proposition 2.15) then converts 'staying $\varepsilon$ away from every maximizer' into a uniform loss of $\theta$ in the growth constant, which is the mechanism behind the concentration theorems.

What would settle it

Simulate the longest convex chain for the exponential density $p(x,y)\propto e^{-4x}$ and for the uniform density on the same triangle at growing $n$, and compare $L_n^{(p)}/L_n^{(unif)}$ with $J_*(p)/2$ computed numerically from the variational formula: the paper predicts convergence to that constant, so a persistent mismatch would refute the claimed density dependence of the growth constant. In the uniform case the companion prediction $(1/n)\log[(3n)!/n!\,P(\text{full chain})]\to\log 54$ can be checked directly against the exact formula $P(\text{full chain})=2^n/(n!(n+1)!)$.

Watch

Extended reading notes

Core claim

At the center of the paper is Theorem 1.1: if $S_n$ is a set of $n$ independent samples from a triangle $T$ with continuous density $p$, and $L_n$ is the length of the longest convex chain in $S_n$, then $L_n/n^{1/3}\to \frac{\alpha}{2}\sup_{f\in\mathcal{F}}J_p(f)$ almost surely and in $L^p$ for every $p\in[1,\infty)$, where $\mathcal{F}$ is the class of continuous convex functions $f:[0,1]\to[0,1]$ with $f(0)=0$, $J_p(f)=\int_0^1 (f''(x)p(x,f(x)))^{1/3}\,dx$, and $\alpha$ is the universal constant of the uniform case established in [4]. The functional $J_p$ is a density-weighted equi-affine arclength, and the supremum $J_*$ is attained because $J$ is upper semicontinuous on a compact metrization of $\mathcal{F}$. Theorem 1.2 states that every convex chain whose length is within $\delta n^{1/3}$ of $L_n$ lies within distance $\varepsilon$ of the maximizer set of $J$, for a $\delta$ depending on the density; Theorem 1.3 proves the same concentration for the conditional law with all $n$ samples forming one convex chain, at a faster exponential rate; and Theorem 1.4 identifies the rate of that rare event as $(1/n)\log[(3n)!/n!\,P(S_n\text{ is a convex chain})]\to \log(27J_*^3/4)$. The growth-rate theorem holds with only continuity of $p$, obtained by perturbing the density with uniform mass; the shape statements assume in addition that $p$ stays bounded away from zero.

Load-bearing premise

The load-bearing premise is that the sampling density stays bounded away from zero everywhere in the triangle: without that, the exponential estimates behind shape concentration, the regularity of maximizers, and the full-chain rare-event rate are not proved, and only the $n^{1/3}$ growth rate itself survives on continuity alone.

Editorial extensions

If this is right

  • The leading-order length of the longest convex chain is $(\alpha/2)J_*(p)\,n^{1/3}$ almost surely: the sampling density changes only the multiplicative constant, not the $n^{1/3}$ scale.
  • Near-longest chains have a deterministic limit shape: any chain of length at least $L_n-\delta n^{1/3}$ is $\varepsilon$-close to the set of maximizers of $J_p$, and when the maximizer is unique the limit is a single explicit curve.
  • The probability that all $n$ points lie in convex position obeys $(1/n)\log[(3n)!/n!\,P(\text{full chain})]\to\log(27J_*^3/4)$, pinning the rare-event rate up to subexponential factors.
  • Conditioning on the full-sample chain changes the regime — fluctuations of $L_n$ vanish — but the same maximizing curves describe the shape, now with concentration at the faster $e^{-cn}$ rate in Theorem 1.3.
  • The functional appearing in the constant is the affine perimeter that already governs limit shapes of random convex lattice polygons, so the result connects random-point chains to that existing universality.

Reading between the lines

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

  • Because the universal constant $\alpha$ is unknown, ratios are the cleanest test of the formula: the theory predicts $L_n^{(p)}/L_n^{(q)}\to J_*(p)/J_*(q)$ for two densities, so taking $q$ uniform (where $J_*=2$) isolates the density dependence that the paper predicts, without needing the value of $\alpha$.
  • The positivity assumption on the density is likely not intrinsic: for densities supported on a proper subtriangle the same proof scheme should give a version of the theorem on that subtriangle, whereas densities that vanish smoothly at the boundary would require a different control of maximizer regularity and of points near the boundary.
  • For separable densities $p(x,y)=u(x)v(y)$ the variational integral may reduce to an Euler–Lagrange equation with explicit solutions, which would supply the first computed non-uniform maximizers and a sharp quantitative test of the shape-concentration statement.
  • The proof bounds fluctuations of $L_n$ by $n^{1/6+o(1)}$ but leaves their true order open; whether the typical fluctuations actually reach the $n^{1/6}$ scale, and with what distribution, is a question the paper explicitly leaves to future work.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 4 minor

Summary. The paper studies the length L_n of the longest convex chain among n i.i.d. points in the triangle T, for a general continuous density p. The main result, Theorem 1.1, states that L_n/n^{1/3} converges almost surely and in every L^p to (alpha/2) sup_{f in F} ∫_0^1 (f''(x) p(x,f(x)))^{1/3} dx, where alpha is the universal constant from the uniform-case theorem of Ambrus and Bárány. The proof splits into a lower bound (Section 4), an upper bound (Section 5), and an approximation argument (Section 6) that removes the strict-positivity assumption (H2) from the growth-rate theorem. The paper also proves shape concentration of near-longest chains around the maximizers of the variational functional (Theorem 1.2), a corresponding conditional statement when all samples form a convex chain (Theorem 1.3), and an asymptotic for the probability of that event (Theorem 1.4).

Significance. If correct, the paper gives the first general-density law of large numbers for longest convex chains, extending the uniform result of Ambrus and Bárány and connecting it to an affine-arclength variational formula analogous to the Deuschel–Zeitouni formula for longest monotone chains. The deterministic analysis of the variational problem is substantial and mostly self-contained: compactness and upper semicontinuity are proved in Section 2, the uniform case is solved explicitly, and the maximizer set is shown to be nonempty and compact. The probabilistic input is also carefully organized around tangency triangles, with external ingredients (the Ambrus–Bárány constant, Valtr's formula, and Talagrand's concentration inequality) cited precisely. The proof of Theorem 1.1 under continuity alone via the epsilon-perturbation in Section 6 is a genuine strength, since it shows that the central growth-rate claim does not depend on the strict-positivity assumption (H2); that assumption is used only for the shape-concentration and rare-event theorems 1.2–1.4. I found no load-bearing error in the central derivation.

minor comments (4)
  1. [Section 4, Lemma 4.2] Lemma 4.2 is stated without proof. Since it supplies the multinomial lower bound used in the proof of Proposition 4.1(b), a citation or a short proof would improve self-containedness; the statement is a standard local central limit theorem and the omission is not a correctness concern.
  2. [Section 5.3, Claim 5.21] The invocation of Proposition 3.6(d) is not literally correct as written: with t = n^{1/13} and beta = 1/4, that proposition bounds a deviation of size n^{1/13} n^{1/4} = n^{17/52}, whereas the display involves the random threshold n^{1/13} N_{n,ell}^{1/4}. The intended estimate follows by conditioning on N_{n,ell} and applying the concentration inequality with s = n^{1/13} N_{n,ell}^{1/4}, together with the high-probability lower bound on N_{n,ell} from Hoeffding; this should be stated explicitly.
  3. [Section 4, proof of Proposition 4.1] There are a few typographical and wording slips: 'we can send delta down to 0 we obtain (4.3)' is missing a 'to', and Claim 5.15 contains 'inequlities'. These do not affect the mathematics.
  4. [Section 5.2, Lemma 5.4] In the proof of Lemma 5.4, 'togther' appears for 'together'. Also, the reduction from S_n to uniform samples U_n via the density bounds could be phrased more cleanly, though the argument is clear.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the uniform-case constant is an external benchmark and the variational formula is derived, not fitted.

full rationale

I walked the derivation chain for Theorem 1.1. The claimed limit is L_n/n^{1/3} -> (alpha/2) sup_f J_p(f), with alpha a universal constant 'not depending on the density function p' that the paper explicitly sources from external prior work: 'The constant α comes from the uniform case p≡2, which was considered by Ambrus and Bárány [4].' This is not a self-citation: [4] is by Ambrus and Bárány, not by Bates and Sen, and the current authors do not appear in the reference list. The variational functional J is defined independently in (1.3) as the integral of (f''(x)p(x,f(x)))^{1/3}, and the uniform-case value J⋆=2 is computed from Proposition 2.1 rather than imposed to match alpha. The observation that the right-hand side of (1.4) reduces to alpha in the uniform case is a consistency check, not a circular reduction, because the uniform-case result itself is imported as an external theorem. The lower bound (Proposition 4.1) and upper bound (Proposition 5.1) are proved separately for general densities via concatenation of tangency triangles and the comparison Proposition 3.6(a), which uses alpha as an external input. I found no fitted parameters, no predictions of fitted quantities, and no equation that makes the target limit equal to its own inputs by construction. The paper honestly notes a technical obstacle in the upper-bound strategy ('we did not manage to accomplish it! Instead, we construct a small number of piecewise convex functions'), but this is a proof-strategy limitation, not circularity. The fragile bounded-below assumption (H2) affects Theorems 1.2–1.4, while Theorem 1.1 is proved under (H1) alone by the approximation argument in Section 6; fragility of an assumption is not circularity. Minor exposition issues, such as Lemma 4.2 being stated without proof and the t=n^{1/13}, β=1/4 invocation in Claim 5.21, do not reduce any result to its inputs. Overall, the derivation is self-contained relative to the advertised external benchmark, and no circular step is present.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No parameters are fitted to data. The universal constant alpha is an input from the uniform-density theorem of Ambrus-Barany; the constants c and C are hypotheses on the density, not fitted quantities. The proof uses technical parameters whose exact powers, such as 1/157 and 1/158, do not affect the conclusions. The main external axioms are cited theorems: the uniform-case constant, Valtr's convex-position formula, and Talagrand's inequality.

assumptions (5)
  • domain assumption The uniform-case constant alpha exists and is finite, with lim E L_unif / n^{1/3} = alpha, as in Ambrus-Barany, Theorem 1.6.
    Invoked in Theorem 1.1 and in Proposition 3.6(a), which calibrates lower and upper bounds; not reproved here. Known bounds are 0.6863 < alpha < 3.4248, with a conjecture that alpha = 3.
  • standard math The probability that n uniform points in a triangle are in convex position with two fixed vertices is 2^n / (n! (n+1)!), attributed to Valtr and Barany-Rote-Steiger-Zhang.
    Used in Proposition 3.6(b) and in Remark 3.8; accepted as a standard external result.
  • standard math Talagrand's convex-distance concentration inequality, stated as Lemma 3.9 and cited from Janson-Luczak-Rucinski, holds for L_n with certificate psi(r) = r.
    Used in Proposition 3.6(d) and Theorem 1.2; this is a standard external concentration theorem.
  • standard math Lemma 4.2, the multinomial lower bound, is stated without proof.
    The paper says 'we omit the proof'; this is a standard local central limit theorem estimate used in the lower bound for full convex chains.
  • domain assumption The density p is continuous (H1) and bounded below by 2c (H2); H1 is required throughout, H2 is required for Theorems 1.2-1.4 and for several positivity arguments.
    These are explicit hypotheses in Section 1.1; the shape-concentration and rare-event claims depend on them.

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Pith. "Pith review of Longest convex chains with i.i.d. points." pith.science (2026). https://pith.science/paper/SVCH3LXS

@misc{pith2026260809105,
  author       = {Pith},
  title        = {Pith review of: Longest convex chains with i.i.d. points},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SVCH3LXS}},
  note         = {Machine review of arXiv:2608.09105}
}
abstract

Sample $n$ i.i.d. points from a triangle, according to some bounded density function. Given two vertices $A,B$ of the triangle, what is the maximum number of samples that form a convex chain with initial point $A$ and terminal point $B$? We show that to leading order, the answer is $cn^{1/3}$, generalizing a result of Ambrus and B\'ar\'any that considered uniformly distributed points. Furthermore, we express the constant $c$ using a variational formula whose maximizer (if unique) gives the limiting curve formed by the longest convex chain. By comparison, for $n$ i.i.d. samples from the unit square, the length of the longest monotone chain is asymptotically $c'n^{1/2}$. Despite the difference in scale, our formula is nicely connected to one established for $c'$ by Deuschel and Zeitouni.

Figures

Figures reproduced from arXiv: 2608.09105 by the authors.

Figure 1
Figure 1. (right) provides an illustration. In this paper, we investigate the longest convex chains within a random collection of points. Namely, we consider n i.i.d. samples from T , and study the longest convex chain found as a subset of these samples. Of primary interest are the length and shape of such a chain. We will show that in the large-n limit, these two quantities satisfy a variational relationship (Theorem 1.1). M… view at source ↗
Figure 2
Figure 2. Heuristic for variational formula (1.4). Left: candidate function f (shown in red) together with its tangency triangle between x and x + ε (shown in blue). Right: a convex chain realized by concatenating smaller chains within the tangency triangles defined by f. Sample points that do not participate in the chain are not shown. of “tangency triangles” of f; see [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Two realizations simulated by Nishant Ajitsaria. Left: n = 25,000 sample points with uniform density p ≡ 2, resulting in Ln = 84. The limit curve given by (1.14) is shown in solid green. Right: n = 25,000 sample points with density p(x, y) = 16 1−5 e−4 e −4x , resulting in Ln = 81. The graph of (1.14) is shown only for comparison [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: A possible choice of the open sets Q1, . . . , Qk, in the case k = 5. Each Qi supports a smooth function φi that is used to place additional mass inside Qi , as in (2.67). By appropriately tuning the mass sizes via parameters λ1, . . . , λk, as in (2.66), we find a den…
Figure 5
Figure 5. Figure 5: The tangency triangle Trif (a, b) is the lightly shaded region on the left. This region contains the smaller tangency triangles Trif (a, x) and Trif (x, b) shown on the right. In both, the graph of f is the solid black curve. Note that R is convex, the points A and B a…
Figure 6
Figure 6. Figure 6: Left: a ρ-regular convex chain. Right: a ρ-irregular convex chain. Lemma 5.8 (Full convex chains are unlikely to be irregular). Assume (H0) and (H2). For every ϵ > 0, there exists ρ = ρ(ϵ, C/c) ∈ (0, 1 2 ) small enough that P(Sn is a ρ-irregular convex chain) ⩽ ϵ n · P…
Figure 7
Figure 7. Figure 7: Argument for Lemma 5.10. Left: The original convex chain Cn, which is assumed to be ρ-irregular. Right: The same chain partitioned into three subsets. The subset in T1 is in convex position with (0, 0) and the new vertex (1−ρ, ρ), shown as a red square. Similarly, the …
Figure 8
Figure 8. Figure 8: Illustration of Step 2b. The piecewise linear function f is shown in solid blue. The shaded region (outlined in dashed blue) is the tangency triangle of f between xℓ and xℓ+1. The vertical coordinates yℓ and yℓ+1 are chosen slightly above f(xℓ) and f(xℓ+1), respectivel…
Figure 9
Figure 9. Figure 9: Illustration of Claim 5.18(a). The elements of Cℓ are shown as solid black circles. Trif (xℓ , xℓ+1) is the shaded region on the left (outlined in dashed blue), while △ℓ = Trif˜(xℓ , xℓ+1) is the shaded region on the right (outlined in solid red). The containment Cℓ ⊂ …

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Pith tools

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