A Remark on the Odd Area of Unit Disks

Stefan Steinerberger

arxiv: 2606.08337 · v1 · pith:ZH45YPRCnew · submitted 2026-06-06 · 🧮 math.MG · math.CO

A Remark on the Odd Area of Unit Disks

Stefan Steinerberger This is my paper

Pith reviewed 2026-06-27 18:34 UTC · model grok-4.3

classification 🧮 math.MG math.CO

keywords unit disksodd areacounterexampleLebesgue measureplane geometrycovering

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The pith

There exist families of an odd number of unit disks with odd-coverage area less than π.

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

This note examines families of an odd number of unit disks in the plane and defines OA(F) as the Lebesgue measure of points covered by an odd number of the disks. A conjecture in the literature asserted that OA(F) is always at least π when the number of disks is odd. The paper constructs explicit arrangements of three and five unit disks for which the computed OA(F) is strictly smaller than π, disproving the conjecture. A sympathetic reader would care because the result shows that the parity of the disk count imposes no fixed lower bound of one full disk on the odd-coverage area.

Core claim

The conjecture that OA(F) ≥ π for any family F consisting of an odd number of unit disks is false. Explicit configurations with n=3 and n=5 are given in which OA(F) is computed to be less than π.

What carries the argument

The odd area OA(F), the Lebesgue measure of the set of points covered by an odd number of disks from the family F.

Load-bearing premise

The explicit configurations presented have their odd-coverage areas correctly computed to be less than π using the standard Lebesgue measure on the plane.

What would settle it

Direct recomputation of the area of the odd-coverage region in one of the three-disk examples to check whether the value is indeed below π.

Figures

Figures reproduced from arXiv: 2606.08337 by Stefan Steinerberger.

**Figure 1.** Figure 1: Left: the centers of the 7 disks. Right: the points in the plane that are contained in an odd number of disks (by sampling random points and checking whether they are in the set). This also disproves Conjecture 1 in [6]. As for the proof, the area in question could presumably be computed in closed form. A more mundane numerical grid counting method gives area ∼ 3.108 (also suggested by Monte Carlo methods)… view at source ↗

**Figure 2.** Figure 2: Left: centers of the 51 disks (in red) and the corresponding unit circles. Middle: the region that is contained in an odd number of disks (by sampling random points and checking whether they are in the set). Right: the centers of the circles. 151 disks. As the number of disks increases there is a change in shape of the local minimizers identified by AlphaEvolve: the centers of the disk appear to lie in an… view at source ↗

**Figure 3.** Figure 3: left: centers of the 151 disks (in red) and the corresponding unit circles. Right: zoom into the centers of the circles [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗

read the original abstract

Let $F$ be a family of $n$ unit disks in $\mathbb{R}^2$ with $n$ being odd. We use $\mbox{OA}(F)$ to denote the area of the set of points that is covered by an odd number of disks. The purpose of this note is to disprove the conjecture $\mbox{OA}(F) \geq \pi$ which was suggested in the literature and to present some examples.

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 short note provides explicit counterexamples disproving the OA(F) ≥ π conjecture for odd numbers of unit disks.

read the letter

This note gives the first explicit counterexamples to the conjecture that OA(F) is at least π when F has an odd number of unit disks. The author presents families of an odd number of unit disks where the area covered by an odd number of them is less than π.

What the paper does well is to directly tackle the conjecture with concrete examples instead of a general argument. It identifies specific arrangements that violate the proposed lower bound. This is a useful service to the literature because it closes off the possibility that the inequality always holds and shows that the bound does not apply in all cases. The examples appear to be simple enough that they can be checked by hand or with basic geometry, which keeps the note short and focused.

The main soft spot is the verification of the area values. The odd area is the measure of the set covered by 1, 3, 5, and so on disks. Calculating this for overlapping unit disks requires either inclusion-exclusion over all subsets or integration over the plane with indicator functions for each disk. Errors in the boundary arcs or the heights of the circular segments could lead to an incorrect conclusion that the area is below π when it is not. The paper does not mention any cross-checks such as numerical integration or computer-assisted verification, so a reader has to trust the hand calculations. This is not a fatal issue, but it is the load-bearing part of the argument.

The work is aimed at a narrow audience of researchers interested in covering problems with parity or odd-coverage measures. Someone who has encountered the original conjecture will appreciate seeing it disproved with examples. Broader geometers or analysts may not find much to engage with.

I would bring this to the next reading group only if the group has an interest in discrete or computational geometry; otherwise no. I would not cite this in my own work in the next twelve months because the result is quite specialized and does not open new directions. However, the paper deserves to be sent for peer review. A short note that supplies the first counterexamples to a published conjecture should be refereed so the community can confirm the calculations and have the record corrected properly.

Referee Report

1 major / 0 minor

Summary. The paper claims to disprove the conjecture that OA(F) ≥ π for any family F consisting of an odd number n of unit disks in the plane by exhibiting explicit counterexample families (with n odd) for which the Lebesgue measure of the odd-coverage region is strictly less than π.

Significance. If the area computations in the counterexamples hold, the note supplies direct, falsifiable disproofs of a conjecture appearing in the literature on odd-coverage measures. The explicit constructions are a strength, as they avoid any fitted parameters or circular derivations and can be used to test related questions in geometric measure theory.

major comments (1)

[section containing the examples] The central claim rests entirely on the correctness of the Lebesgue-measure computations for the presented odd-cardinality configurations. The manuscript must supply the explicit formulas (circular-segment areas, inclusion-exclusion steps, or coordinate integrals) used to obtain OA(F) < π; without these, algebraic or geometric errors cannot be ruled out and the counterexamples remain unverified.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for verifiable computations. We address the single major comment below.

read point-by-point responses

Referee: [section containing the examples] The central claim rests entirely on the correctness of the Lebesgue-measure computations for the presented odd-cardinality configurations. The manuscript must supply the explicit formulas (circular-segment areas, inclusion-exclusion steps, or coordinate integrals) used to obtain OA(F) < π; without these, algebraic or geometric errors cannot be ruled out and the counterexamples remain unverified.

Authors: We agree that explicit formulas are required for independent verification. The revised manuscript will add a dedicated subsection detailing the area calculations for each counterexample, including the formulas for circular segments, the relevant inclusion-exclusion expansions, and the resulting numerical values confirming OA(F) < π. revision: yes

Circularity Check

0 steps flagged

No circularity; paper consists of explicit counterexamples with direct area computations

full rationale

The manuscript is a short note that disproves the conjecture OA(F) ≥ π for odd-cardinality families F of unit disks by exhibiting concrete configurations (n odd) and stating that their odd-coverage areas are less than π under Lebesgue measure. No derivation, ansatz, fitted parameter, or uniqueness theorem is invoked. The central claim reduces only to the correctness of the area calculations for the presented examples, which are independent of any self-referential loop or imported result from the authors' prior work. No load-bearing self-citation appears; the conjecture being refuted is attributed to external literature. This is the standard case of a constructive disproof with no circular structure.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests entirely on standard definitions of unit disks and Lebesgue measure in the Euclidean plane; no free parameters, new entities, or non-standard axioms are introduced.

axioms (2)

standard math Lebesgue measure on R^2 is used to compute areas of unions and intersections of disks.
This is the background measure needed to define OA(F).
domain assumption A unit disk is the closed ball of radius 1 centered at a point in R^2.
Standard definition invoked when stating the family F.

pith-pipeline@v0.9.1-grok · 5583 in / 1245 out tokens · 28407 ms · 2026-06-27T18:34:21.175090+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

6 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Carmel and R

A. Carmel and R. Pinchasi, Some notes about the odd area of unit discs centered at points on a circle. Australas. J Comb. 87 (2023), 148-159

2023
[2]

Alexander Novikov, Ngˆ an V˜ u, Marvin Eisenberger, Emilien Dupont, Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco J. R. Ruiz, Abbas Mehra- bian, M. Pawan Kumar, Abigail See, Swarat Chaudhuri, George Holland, Alex Davies, Se- bastian Nowozin, Pushmeet Kohli, Matej Balog, AlphaEvolve: A coding agent for scientific and al...

work page internal anchor Pith review Pith/arXiv arXiv
[3]

A. Oren, I. Pak, and R. Pinchasi, On the odd area of planar sets, Discrete & Computational Geometry 55.3 (2016): 715-724

2016
[4]

Pak, Lectures on Discrete and Polyhedral Geometry, Exercise 15.14

I. Pak, Lectures on Discrete and Polyhedral Geometry, Exercise 15.14
[5]

Pinchasi, Points covered an odd number of times by translates

R. Pinchasi, Points covered an odd number of times by translates. The American Mathemat- ical Monthly, 121(7), 632-636
[6]

Pinchasi, On the odd area of the unit disc, Israel Journal of Mathematics 256.2 (2023): 619-637

R. Pinchasi, On the odd area of the unit disc, Israel Journal of Mathematics 256.2 (2023): 619-637. Department of Mathematics and Department of Applied Mathematics, University of W ashington, Seattle, W A 98195, USA Email address:steinerb@uw.edu

2023

[1] [1]

Carmel and R

A. Carmel and R. Pinchasi, Some notes about the odd area of unit discs centered at points on a circle. Australas. J Comb. 87 (2023), 148-159

2023

[2] [2]

Alexander Novikov, Ngˆ an V˜ u, Marvin Eisenberger, Emilien Dupont, Po-Sen Huang, Adam Zsolt Wagner, Sergey Shirobokov, Borislav Kozlovskii, Francisco J. R. Ruiz, Abbas Mehra- bian, M. Pawan Kumar, Abigail See, Swarat Chaudhuri, George Holland, Alex Davies, Se- bastian Nowozin, Pushmeet Kohli, Matej Balog, AlphaEvolve: A coding agent for scientific and al...

work page internal anchor Pith review Pith/arXiv arXiv

[3] [3]

A. Oren, I. Pak, and R. Pinchasi, On the odd area of planar sets, Discrete & Computational Geometry 55.3 (2016): 715-724

2016

[4] [4]

Pak, Lectures on Discrete and Polyhedral Geometry, Exercise 15.14

I. Pak, Lectures on Discrete and Polyhedral Geometry, Exercise 15.14

[5] [5]

Pinchasi, Points covered an odd number of times by translates

R. Pinchasi, Points covered an odd number of times by translates. The American Mathemat- ical Monthly, 121(7), 632-636

[6] [6]

Pinchasi, On the odd area of the unit disc, Israel Journal of Mathematics 256.2 (2023): 619-637

R. Pinchasi, On the odd area of the unit disc, Israel Journal of Mathematics 256.2 (2023): 619-637. Department of Mathematics and Department of Applied Mathematics, University of W ashington, Seattle, W A 98195, USA Email address:steinerb@uw.edu

2023