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arxiv: 2310.18143 · v1 · submitted 2023-10-27 · 🧮 math.MG · math.PR

A central limit theorem for random disc-polygons in smooth convex discs

Ferenc Fodor , D\'aniel I. Papv\'ari This is my paper

Pith reviewed 2026-05-24 06:37 UTC · model grok-4.3

classification 🧮 math.MG math.PR
keywords central limit theoremStein's methodrandom disc-polygonsconvex geometryarea functionalvariance boundsmooth convex discsC^2 boundary
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The pith

The area of uniform random disc-polygons in smooth convex discs obeys a quantitative central limit theorem.

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

This paper proves a quantitative central limit theorem for the area of uniform random disc-polygons inside smooth convex discs with C^2_+ boundaries. It combines Stein's method with a previously established asymptotic lower bound on the variance of the area. A reader would care because this describes the asymptotic distribution of the area fluctuations in these random geometric constructions. The result extends earlier work on variance to full normality with error bounds. It applies specifically when the boundary is twice differentiable with positive curvature.

Core claim

We prove a quantitative central limit theorem for the area of uniform random disc-polygons in smooth convex discs whose boundary is C^2_+. We use Stein's method and the asymptotic lower bound for the variance of the area proved by Fodor, Grünfelder and Vigh (2022).

What carries the argument

Stein's method for normal approximation applied to the area of random disc-polygons, using the variance lower bound as input.

If this is right

  • The normalized area of the random disc-polygon converges in distribution to a standard normal random variable.
  • Explicit quantitative bounds are obtained on the distance between the distribution of the normalized area and the normal distribution.
  • The central limit theorem applies to every smooth convex disc whose boundary satisfies the C^2_+ condition.
  • The proof depends on the variance of the area growing according to the asymptotic lower bound established earlier.

Where Pith is reading between the lines

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

  • The same Stein-method approach might yield central limit theorems for other additive functionals of the random disc-polygon such as its perimeter.
  • If the variance lower bound is asymptotically sharp, the rate of convergence in the central limit theorem could be optimal for this model.
  • The result suggests that similar quantitative normality statements could hold for random polytopes generated by other smooth convex bodies in the plane.

Load-bearing premise

The asymptotic lower bound on the variance of the area from the 2022 paper continues to hold under the C^2_+ boundary condition.

What would settle it

A computation or simulation for a specific C^2_+ convex disc showing that the normalized area of random disc-polygons deviates from normality or that the variance fails to satisfy the stated lower bound.

read the original abstract

In this paper we prove a quantitative central limit theorem for the area of uniform random disc-polygons in smooth convex discs whose boundary is $C^2_+$. We use Stein's method and the asymptotic lower bound for the variance of the area proved by Fodor, Gr\"unfelder and V\'igh (2022).

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 proves a quantitative central limit theorem for the area of uniform random disc-polygons in smooth convex discs whose boundary is C^2_+. It applies Stein's method to obtain a rate of convergence in the CLT, relying on the asymptotic lower bound for the variance of the area established in Fodor, Grünfelder and Vigh (2022).

Significance. If the result holds, the paper supplies a rate for the CLT in the setting of random disc-polygons, extending the literature on fluctuations of functionals of random convex bodies. The combination of Stein's method with the external variance bound is a standard and efficient route; credit is due for correctly reducing the CLT derivation to the prior variance estimate under the stated C^2_+ hypotheses.

major comments (1)
  1. [§1] The central claim depends on the variance lower bound from Fodor et al. (2022); §1 and the statement of the main theorem should contain an explicit paragraph verifying that the C^2_+ smoothness and the uniform random disc-polygon construction satisfy every hypothesis required by that 2022 result, including any implicit constants in the geometric estimates.
minor comments (2)
  1. Notation for the disc-polygon and the underlying convex body should be introduced once in a dedicated notation paragraph rather than piecemeal.
  2. [Abstract] The abstract states the result but does not record the explicit rate obtained; adding the rate (even in O-notation) would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation and recommendation for minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: The central claim depends on the variance lower bound from Fodor et al. (2022); §1 and the statement of the main theorem should contain an explicit paragraph verifying that the C^2_+ smoothness and the uniform random disc-polygon construction satisfy every hypothesis required by that 2022 result, including any implicit constants in the geometric estimates.

    Authors: We agree that an explicit verification paragraph will improve clarity. In the revised manuscript we will add a dedicated paragraph in §1 immediately following the statement of the main theorem. This paragraph will list each hypothesis of Fodor, Grünfelder and Vigh (2022) and confirm that the C^2_+ boundary assumption together with the uniform random disc-polygon model satisfy them all, including the geometric estimates and any implicit constants appearing in their variance asymptotics. The paragraph will reference the relevant statements from the 2022 paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation applies Stein's method (an external, standard technique) to obtain a quantitative CLT for the area functional, taking as given only the asymptotic variance lower bound established in the separate 2022 paper by Fodor, Grünfelder and Vigh. No equation or step within the present manuscript reduces the claimed CLT result to a self-definition, a fitted parameter renamed as prediction, or any other enumerated circular pattern; the cited variance bound is treated as an independent external input rather than derived or assumed within this work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard mathematical background (probability theory, Stein's method) and the external 2022 variance bound; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The boundary of the convex disc is C^2_+
    Invoked as the necessary smoothness condition for the theorem.

pith-pipeline@v0.9.0 · 5572 in / 1249 out tokens · 26558 ms · 2026-05-24T06:37:40.102876+00:00 · methodology

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Reference graph

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