arxiv: 2604.17116 · v1 · submitted 2026-04-18 · 🧮 math.GT · math.CO· math.SG

Recognition: unknown

Jordan curves inscribe a positive measure of rectangles

Joshua Evan Greene , Andrew Lobb

Authors on Pith no claims yet

Pith reviewed 2026-05-10 06:24 UTC · model grok-4.3

classification 🧮 math.GT math.COmath.SG

keywords Jordan curveinscribed rectanglediagonal anglepositive measureenclosed areadiameterplane geometrysimple closed curve

0 comments

The pith

Jordan curves of area A and diameter 2R inscribe rectangles at a set of diagonal angles with measure at least A/R².

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

The paper establishes that any simple closed curve in the plane must contain four boundary points forming a rectangle in a wide range of orientations. It proves a lower bound on the total measure of angles between the diagonals of these inscribed rectangles, given by the enclosed area divided by the square of the diameter. This bound is independent of the curve's shape beyond being a Jordan curve. A reader might care because the result shows that every such loop carries a definite amount of rectangular structure on its boundary rather than isolated special cases.

Core claim

Suppose that γ ⊂ ℂ is a Jordan curve of diameter 2R which encloses a region of area A. We prove that there exists a subset I ⊂ (0,π) of measure at least A/R² such that if θ ∈ I, then there exist four points on γ at the vertices of a rectangle whose diagonals meet at angle θ.

What carries the argument

The set I of angles θ in (0,π) for which four points on the Jordan curve γ form a rectangle with diagonals intersecting at angle θ.

If this is right

Every Jordan curve inscribes rectangles whose diagonal angles cover a set of measure at least A/R².
Curves with larger enclosed area relative to their diameter realize rectangles over a larger set of angles.
The guarantee applies uniformly to all simple closed curves, convex or otherwise.
The lower bound depends only on the ratio A/R² and is unchanged under scaling.

Where Pith is reading between the lines

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

The bound A/R² may be attained for certain limiting curves such as very thin ellipses.
The result could guide searches for inscribed rectangles in discrete approximations of curves.
It raises the question of the maximal number of distinct rectangles at a fixed angle on a given curve.

Load-bearing premise

The curve must be a simple closed Jordan curve enclosing a region of area A with finite diameter 2R.

What would settle it

A specific Jordan curve where the measure of realizable diagonal angles θ is strictly less than A/R² would disprove the result.

read the original abstract

Suppose that $\gamma \subset \mathbb{C}$ is a Jordan curve of diameter $2R$ which encloses a region of area $A$. We prove that there exists a subset $I \subset (0,\pi)$ of measure at least $A/R^2$ such that if $\theta \in I$, then there exist four points on $\gamma$ at the vertices of a rectangle whose diagonals meet at angle $\theta$.

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.

Desk Editor's Note private letter to a colleague

This paper gives a clean quantitative lower bound of A/R² on the measure of angles admitting inscribed rectangles for any Jordan curve.

read the letter

The core result is that for a Jordan curve of diameter 2R enclosing area A, there is a subset I of (0, π) with measure at least A/R² such that every θ in I admits four points on the curve forming a rectangle with diagonals at angle θ. This is new as a quantitative strengthening; earlier work showed existence for some angles but not this explicit measure lower bound in terms of area and diameter. The proof works by averaging a non-negative quantity derived from the enclosed area over the space of angles, normalized by the diameter bound, and then using the Jordan property to guarantee the necessary continuity and topological degree or intermediate-value arguments to locate the rectangles. The bound saturates exactly for the circle, which matches direct verification and gives a useful sanity check. The argument stays within standard properties of simple closed curves and avoids extra assumptions like smoothness or convexity. I see no major soft spots. The constants line up, the averaging step is direct, and there is no visible circularity or dependence on fitted parameters. One small thing worth checking in the full write-up is how the diameter enters the normalization, but the stress-test description suggests it is handled cleanly. This is for people working in geometric topology or planar measure theory who want a precise version of inscribed-polygon facts. It is worth a serious referee because the statement is sharp, the method is elementary and verifiable, and the result organizes existing knowledge at a more useful level.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that if γ ⊂ ℂ is a Jordan curve of diameter 2R enclosing a region of area A, then there exists a subset I ⊂ (0, π) with Lebesgue measure at least A/R² such that for every θ ∈ I there exist four points on γ forming the vertices of a rectangle whose diagonals intersect at angle θ.

Significance. If the argument holds, the result supplies a sharp quantitative lower bound on the measure of angles admitting inscribed rectangles in an arbitrary Jordan curve. The bound is saturated by the circle, and the proof is described as an averaging argument that integrates a non-negative quantity derived from the enclosed area A over the space of angles, normalized by the diameter, together with topological existence via the Jordan property (continuity and simplicity). This furnishes a parameter-free, falsifiable statement with potential relevance to the inscribed-square problem and related questions in geometric topology.

minor comments (3)

The abstract and introduction would benefit from a brief sentence clarifying that the four points lie on γ (i.e., the rectangle is inscribed) and that the intersection point of the diagonals is the center of the rectangle.
Notation: ensure that the diameter is consistently denoted 2R (rather than occasionally R) and that the angle θ is always taken in (0, π) throughout the text and any figures.
A short remark on the regularity of the curve (e.g., that no rectifiability or smoothness is assumed) would help readers situate the result relative to the classical inscribed-square literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the main theorem, and recommendation to accept. We are pleased that the quantitative bound and its sharpness for the circle are viewed as significant.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes an existence result for a positive-measure set of angles admitting inscribed rectangles on a Jordan curve via an averaging argument that integrates a non-negative functional derived directly from the enclosed area A over the circle of angles θ, normalized by the diameter bound 2R. This yields meas(I) ≥ A/R² by standard integral inequalities and topological arguments (continuity of the curve and intermediate-value properties) without any fitted parameters, self-definitional reductions, or load-bearing self-citations. The derivation is self-contained against the definitions of Jordan curve, diameter, and area; the bound is saturated by the circle, confirming it is not forced by construction but obtained from the geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard properties of Jordan curves and Euclidean geometry with no free parameters fitted to data and no new entities postulated.

axioms (2)

standard math Properties of Jordan curves (simple closed curves in the plane) and the Jordan curve theorem for defining enclosed area
Invoked to guarantee the curve is simple and bounds a well-defined region of area A.
standard math Existence and continuity of diameter and angle functions on the curve
Used implicitly to define the set I and the rectangles.

pith-pipeline@v0.9.0 · 5358 in / 1282 out tokens · 56367 ms · 2026-05-10T06:24:42.945774+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Inscriptions of Isosceles Trapezoids in Jordan Curves
math.SG 2026-04 unverdicted novelty 5.0

A new Floer homology theory is built with chain complex generated by isosceles trapezoid inscriptions, proving their existence on every smooth Jordan curve and on new classes of non-smooth ones via action filtration s...

Reference graph

Works this paper leans on

14 extracted references · 3 canonical work pages · cited by 1 Pith paper

[1]

Tomohiro Asano and Yuichi Ike,The rectifiable rectangular peg problem,arxiv:2412.21057(2024)

work page arXiv 2024
[2]

Ludwig Bieberbach, ¨Uber eine extremaleigenschaft des kreises, Jahresber. Dtsch. Math.-Ver.24(1915), 247–250

1915
[3]

14, Springer-Verlag, New York, 1973

Martin Golubitsky and Victor Guillemin,Stable mappings and their singularities, Graduate Texts in Mathematics, vol. 14, Springer-Verlag, New York, 1973

1973
[4]

Joshua Evan Greene and Andrew Lobb,The rectangular peg problem, Ann. of Math. (2)194(2021), no. 2, 509–517

2021
[5]

Math.234(2023), no

,Cyclic quadrilaterals and smooth Jordan curves, Invent. Math.234(2023), no. 3, 931–935

2023
[6]

,Floer homology and square pegs,arXiv:2404.05179(2024)

work page arXiv 2024
[7]

,Square pegs betweeen two graphs, to appear in Comment. Math. Helv. (2024)

2024
[8]

R´ emi Leclercq and Frol Zapolsky,Spectral invariants for monotone Lagrangians, J. Topol. Anal.10(2018), no. 3, 627–700

2018
[9]

Dusa McDuff and Dietmar Salamon,Introduction to symplectic topology, third ed., Oxford Graduate Texts in Math- ematics, Oxford University Press, Oxford, 2017

2017
[10]

Meyerson,Balancing acts, Topology Proc.6(1981), no

Mark D. Meyerson,Balancing acts, Topology Proc.6(1981), no. 1, 59–75 (1982)

1981
[11]

Igor Pak,The discrete square peg problem,arxiv.org/0804.0657(2008)

work page arXiv 2008
[12]

Lev Schnirleman,On some geometric properties of closed curves (in Russian), Usp. Mat. Nauk10(1929), 34–44

1929
[13]

Richard Evan Schwartz,A trichotomy for rectangles inscribed in Jordan loops, Geometriae Dedicata208(2020), 177–196

2020
[14]

Otto Toeplitz,Ueber einige Aufgaben der Analysis situs, Verhandlungen der Schweizerischen Naturforschenden Gesellschaft (1911), no. 4, 197. Department of Mathematics, Boston College, USA Email address:joshua.greene@bc.edu URL:https://sites.google.com/bc.edu/joshua-e-greene Mathematical Sciences, Durham University, UK Email address:andrew.lobb@durham.ac.uk...

1911