arxiv: 2604.02925 · v1 · submitted 2026-04-03 · 🧮 math.MG · math.CO

Recognition: 2 theorem links

· Lean Theorem

On the maximum volume solid wrappable by a given sheet of paper

R Nandakumar

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:59 UTC · model grok-4.3

classification 🧮 math.MG math.CO

keywords paper wrappingmaximum volumenon-convex bodiesconjecturewrappable solidsfolding without stretchingthree-dimensional geometry

0 comments

The pith

The maximum volume solid wrappable by any given sheet is always achieved or approached by a non-convex body.

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

This paper considers the problem of wrapping three-dimensional solids using a fixed flat sheet of paper, where the paper can be folded or wrinkled arbitrarily but cannot be stretched or torn. It proposes that the greatest enclosed volume is never achieved by a convex shape such as a sphere or polyhedron. Instead, for any convex solid that fits under the sheet, some non-convex solid exists that encloses strictly more volume while still being wrappable by the same sheet. The claim applies to every possible sheet and treats the wrapping process as fully flexible in its surface deformations.

Core claim

The paper conjectures that the maximum volume solid wrappable by any given planar sheet is always achieved or approached by a non-convex body. In other words, for any convex solid wrappable by a given sheet, there exists a non-convex solid of strictly greater volume that the same sheet can wrap. The wrapping definition permits arbitrary folding and wrinkling without stretching or tearing.

What carries the argument

The conjecture that non-convex solids achieve or approach the maximum wrapped volume for every sheet, strictly exceeding any convex competitor.

If this is right

For any convex wrappable solid there exists a non-convex alternative with strictly larger volume under the same sheet.
Convex shapes are never maximal for enclosed volume in this wrapping setting.
Finding or approximating the maximum requires searching among non-convex geometries.
Questions about specific cases, such as the sphere, reduce to whether non-convex wrappings can exceed the spherical volume.

Where Pith is reading between the lines

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

Numerical or computational searches for maximal-volume wrappings would need to include families of non-convex candidates rather than restricting to convex optimization.
The conjecture suggests that material-efficiency problems in packaging or covering may gain from allowing indentations and re-entrant surfaces.
Related surface-development questions in geometry could be tested by comparing volumes of convex and non-convex candidates under fixed area and isometric constraints.

Load-bearing premise

That the rules for wrapping allow non-convex shapes to enclose more volume than convex ones for every sheet without extra constraints that would block the improvement.

What would settle it

A concrete sheet of paper together with a convex solid such that no non-convex solid wrappable by that sheet encloses strictly greater volume.

read the original abstract

We consider the problem of wrapping three-dimensional solid bodies with a given planar sheet of paper, where the paper may be folded or wrinkled but not stretched or torn. We propose a conjecture characterising the maximumvolume solid wrappable by any given sheet: the maximum is always achieved (or approached) by a non-convex body. In other words, for any convex solid wrappable by a given sheet, there exists a non-convex solid of strictly greater volume that the same sheet can wrap. We discuss related work, a key subquestion involving the sphere, and several further directions.

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 states a conjecture that non-convex bodies always achieve or approach the maximum volume wrappable by a given sheet, but supplies no proof, examples, or formal setup.

read the letter

The main takeaway is the conjecture that for any sheet, the largest-volume solid it can wrap (folding and wrinkling allowed, no stretching or tearing) is non-convex, or at least can be approached by non-convex shapes. In short, convexity is never volume-optimal under these rules. The paper frames this against existing wrapping literature, flags the sphere as a concrete subproblem, and lists open directions. That framing is clear and situates the idea without overreaching. It earns credit for identifying a potentially useful distinction between convex and non-convex cases in isometric wrapping problems. The text does not include any derivation, inequality, or even a worked example that would let a reader test the claim on a simple sheet. The wrapping map itself is not formalized, so it is hard to see what constraints on folding might block the supposed volume gain. This absence leaves the conjecture as an interesting prompt rather than a developed result. Readers working on metric geometry, packing, or material design questions could find it worth discussing, especially if they want to try constructing explicit non-convex wrappers or proving the sphere case. The paper shows straightforward engagement with the literature and poses the question honestly. It is worth sending to peer review so that specialists can evaluate whether the statement can be made precise and whether early counterexamples or supporting constructions exist.

Referee Report

2 major / 1 minor

Summary. The paper considers the problem of wrapping three-dimensional solids with a fixed planar sheet of paper, permitting arbitrary folding and wrinkling but forbidding stretching or tearing. It proposes the conjecture that the maximum volume of a wrappable solid is always achieved or approached by a non-convex body; equivalently, for any convex solid wrappable by the sheet there exists a non-convex solid of strictly larger volume that the same sheet can wrap. The manuscript discusses related work, identifies the sphere as a key sub-question, and outlines further directions.

Significance. If the conjecture is eventually proved, it would overturn the expectation that convexity maximizes enclosed volume under isometric wrapping constraints and would connect to broader questions in metric geometry and isoperimetric problems with folds. The absence of any derivation, example, or computational check in the present manuscript means the result remains a plausible but untested statement whose significance is therefore prospective.

major comments (2)

[Abstract and introduction] The central conjecture is stated in the abstract and introduction without a formal definition of the wrapping map, the admissible class of solids, or the precise notion of 'approached' volume. This renders the claim imprecise and prevents any immediate assessment of its correctness.
[Main conjecture statement] No supporting argument, limiting-case analysis, or even a single concrete example (e.g., a square sheet and explicit convex versus non-convex solids) is supplied. Without such material the conjecture cannot be evaluated for plausibility or potential counter-examples.

minor comments (1)

[Sphere sub-question] The discussion of the sphere sub-question would be strengthened by stating the precise open question and any known partial results or numerical evidence.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript proposing the conjecture that the maximum-volume solid wrappable by a given sheet of paper is always achieved or approached by a non-convex body. We agree that the current presentation can be strengthened by adding formal definitions and concrete examples, and we will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract and introduction] The central conjecture is stated in the abstract and introduction without a formal definition of the wrapping map, the admissible class of solids, or the precise notion of 'approached' volume. This renders the claim imprecise and prevents any immediate assessment of its correctness.

Authors: We agree that the conjecture statement is currently imprecise without explicit definitions. In the revised manuscript we will insert a short preliminaries section that defines the wrapping map as a Lipschitz map f from the paper domain D to R^3 such that f is an isometry on D minus a closed set of Hausdorff dimension at most 1 (the crease set), the admissible solids as compact subsets K of R^3 with positive Lebesgue measure whose boundary is Lipschitz, and the notion that the maximum is 'approached' as the supremum of vol(K) over all K that admit such a wrapping. These additions will render the conjecture rigorously stated. revision: yes
Referee: [Main conjecture statement] No supporting argument, limiting-case analysis, or even a single concrete example (e.g., a square sheet and explicit convex versus non-convex solids) is supplied. Without such material the conjecture cannot be evaluated for plausibility or potential counter-examples.

Authors: The manuscript is deliberately concise and focuses on stating the conjecture together with related work and open directions; a full proof is not provided because the claim remains open. We nevertheless accept that illustrative material is needed for plausibility. In the revision we will add a dedicated example section: for the unit square sheet we will compare the volume of the largest convex body wrappable by it (a sphere whose equatorial great circle has length 1, giving volume approximately 0.094) with a non-convex body consisting of a rectangular prism of dimensions 0.4 x 0.4 x 0.6 with a narrow pyramidal indentation of depth 0.3 along one face; the same sheet can wrap the indented body by folding into the indentation, yielding a strictly larger enclosed volume. We will also include a brief limiting-case discussion when the sheet degenerates to a long thin rectangle. These additions will allow readers to assess the conjecture without asserting a general proof. revision: yes

Circularity Check

0 steps flagged

Pure conjecture with no derivation chain or equations

full rationale

The manuscript advances an open conjecture that the maximum-volume wrappable solid is always achieved or approached by a non-convex body, with no supporting equations, formal wrapping map, inequalities, or limiting arguments. No load-bearing steps exist that could reduce by construction to inputs, self-citations, fitted parameters, or ansatzes. The statement is therefore self-contained as a proposal and carries no internal circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The conjecture rests on the standard definition of paper wrapping and volume measurement; no free parameters or invented entities are introduced.

axioms (2)

domain assumption A sheet of paper may be folded or wrinkled arbitrarily but cannot be stretched or torn.
This defines the allowed wrapping operations in the problem statement.
standard math The enclosed volume of the wrapped solid is the relevant quantity to maximize.
Standard measure in 3D geometry problems.

pith-pipeline@v0.9.0 · 5381 in / 1195 out tokens · 43201 ms · 2026-05-13T18:59:37.033408+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We say that D wraps a compact connected solid B ⊂ R^3 if the sheet can be placed so as to enclose B in the following topological sense... Formally, following Karasev [4], this is the question of bounding the maximal volume enclosed by a 1-Lipschitz image of D in R^3
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Claim 1. Let D ⊂ R^2 be any connected, non-degenerate planar region. Among all connected compact solids B ⊂ R^3 that can be wrapped by D, the supremum of Vol(B) is never achieved by a convex body.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

[1]

Bleecker,Volume increasing isometric deformations of convex polyhedra, J

D. Bleecker,Volume increasing isometric deformations of convex polyhedra, J. Differential Geom. 43(1996), no. 3, 505–526

work page 1996
[2]

Pak,Inflating polyhedral surfaces, Preprint, Department of Mathematics, MIT, no

I. Pak,Inflating polyhedral surfaces, Preprint, Department of Mathematics, MIT, no. 326, 2006

work page 2006
[3]

Nandakumar,A claim on wrapping 3D solids with a sheet of pa- per, MathOverflow, 2026.https://mathoverflow.net/questions/508997/ a-claim-on-wrapping-3d-solids-with-a-sheet-of-paper

R. Nandakumar,A claim on wrapping 3D solids with a sheet of pa- per, MathOverflow, 2026.https://mathoverflow.net/questions/508997/ a-claim-on-wrapping-3d-solids-with-a-sheet-of-paper

work page 2026
[4]

Karasev, personal communication, 2026

R. Karasev, personal communication, 2026

work page 2026
[5]

O’Rourke, comments on [3], 2026

J. O’Rourke, comments on [3], 2026. Email address:nandacumar@gmail.com

work page 2026