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arxiv: 2110.12191 · v1 · submitted 2021-10-23 · 🧮 math.MG · math.PR

On random disc-polygons in a disc-polygon

Ferenc Fodor , P\'eter Kevei , Viktor V\'igh This is my paper

Pith reviewed 2026-05-24 13:11 UTC · model grok-4.3

classification 🧮 math.MG math.PR
keywords random disc-polygonsconvex disc-polygonsasymptotic formulasvertex numbermissed arear-convexityintegral geometry
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The pith

The expected vertex count and missed area of random disc-polygons inside convex disc-polygons obey the same asymptotic formulas as classical random polygons inside convex polygons.

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

This paper establishes asymptotic formulas for the expected number of vertices and the expected missed area of a uniform random disc-polygon chosen inside a fixed convex disc-polygon. The results are direct analogues of the 1964 theorems of Rényi and Sulanke for ordinary random polygons inside convex polygons. A sympathetic reader cares because the work shows that the probabilistic geometry of random convex sets extends from strict convexity to the r-convex setting without changing the leading-order behavior. The proofs adapt the original integral-geometric limit arguments to the disc-polygon case.

Core claim

For a convex disc-polygon K with n sides, the expected number of vertices of a uniform random disc-polygon inside K and the expected area of K minus that random disc-polygon both admit asymptotic expansions as n tends to infinity that match the corresponding expansions known for ordinary convex polygons.

What carries the argument

Uniform random disc-polygon inscribed in a fixed convex disc-polygon, with the vertex-count and area functionals analyzed via adapted limit arguments from integral geometry.

If this is right

  • The expected vertex number of the random disc-polygon tends to infinity at the same rate as in the polygonal case.
  • The expected missed area tends to zero at the same rate as in the polygonal case.
  • The same probabilistic limit laws for the normalized functionals hold under the disc-polygon model.
  • The results apply to any convex disc-polygon container, not just to circles or smooth bodies.

Where Pith is reading between the lines

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

  • The same proof strategy may extend to other r-convex bodies whose boundaries consist of circular arcs.
  • Numerical sampling of random disc-polygons could provide independent verification of the derived constants in the asymptotics.
  • The work suggests that many other functionals of random convex sets may be insensitive to the precise notion of convexity used.

Load-bearing premise

The limit arguments and integral-geometric techniques developed for ordinary convex polygons carry over to convex disc-polygons after only minor modifications.

What would settle it

An explicit calculation, for a fixed convex disc-polygon with increasing numbers of sides, of the exact expectations that yields leading terms different from those in the classical Rényi-Sulanke formulas.

Figures

Figures reproduced from arXiv: 2110.12191 by Ferenc Fodor, P\'eter Kevei, Viktor V\'igh.

Figure 1
Figure 1. Figure 1: Proof. Let x0 be the vertex of D(u, t) = P \ (B◦ + p), and assume that ∂B + p intersects ∂P in a and b, consequently `(u, t) > |ab| ≥ 2`(u, t)/π. First we prove the lower bound. Draw a line f through x0 that is perpendicular to ab, let z = f ∩ ab, and w.l.o.g. assume |z − a| ≥ |z − b|. Denote by y the intersection point of f and ∂B + p, see [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Computing `1 Assume that for a sufficiently small t the cap D(u, t) = P \ (B◦ + p) contains a single vertex v of P, and denote by e end e ∗ the two edges of P that meet at v. Let c be the centre of the unit circle that determines e, and n = v − c. The circle S 1 + p intersects S 1 + c in y, and the segment pv in z, cf [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original abstract

We prove asymptotic formulas for the expectation of the vertex number and missed area of uniform random disc-polygons in convex disc-polygons. Our statements are the $r$-convex analogues of the classical results of R\'enyi and Sulanke (1964) about random polygons in convex polygons.

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

0 major / 3 minor

Summary. The manuscript proves asymptotic formulas for the expected number of vertices and the expected missed area of a uniform random disc-polygon generated by n points inside a fixed convex disc-polygon K, as n tends to infinity. These are presented as the direct r-convex analogues of the classical Rényi–Sulanke (1964) results for ordinary convex polygons.

Significance. If the derivations are correct, the paper supplies a clean extension of the 1964 integral-geometric arguments to convex bodies whose boundary consists of finitely many circular arcs of constant curvature. The leading asymptotics (log n for vertices, 1/n for missed area) are shown to persist after the natural case split between arc interiors and vertex neighborhoods, with only local modifications to the cap probabilities. This is a modest but useful incremental contribution to the theory of random polytopes in piecewise-C² convex bodies.

minor comments (3)
  1. [§2] §2: the definition of a disc-polygon and the precise meaning of “uniform random disc-polygon” should be stated explicitly before the main theorems, rather than being left implicit from the Rényi–Sulanke reference.
  2. [Main theorems] The statements of the main theorems (presumably Theorems 1 and 2) should record the dependence of the leading constants on the number and curvatures of the boundary arcs of K; the current abstract leaves this dependence unstated.
  3. [Introduction] A short paragraph comparing the new constants with the classical polygonal case would help readers assess the size of the “minor modifications” claimed in the introduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work as a clean r-convex analogue of the Rényi–Sulanke results and for the recommendation of minor revision. No specific major comments appear in the report, so there are no individual points requiring point-by-point rebuttal or manuscript changes.

Circularity Check

0 steps flagged

No significant circularity; explicit extension of external 1964 results

full rationale

The paper states its results are the r-convex analogues of Rényi-Sulanke (1964) and claims only minor modifications to those external integral-geometric arguments are needed when the ambient body is a disc-polygon. The load-bearing step is the direct carry-over of cap-probability calculations after replacing straight sides by circular arcs; this is an external reference, not a self-citation chain or fitted input renamed as prediction. No self-definitional steps, ansatz smuggling, or renaming of known results appear. The derivation therefore remains self-contained against the cited 1964 benchmark.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities are stated. The result rests on standard measure-theoretic probability and integral geometry assumptions imported from prior literature.

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Reference graph

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