Existence of a non-standard isoperimetric triple partition

Emanuele Paolini; Matteo Novaga; Vincenzo Maria Tortorelli

REVIEW 2 major objections 4 minor 1 cited by

Existence of a non-standard isoperimetric triple partition

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read An isoperimetric 3-partition of $\mathbb{R}^8$ exists that is not a lens: its blow-down is a singular minimal cone.

desk verdict A likely-correct resolution of a real open problem, with one load-bearing step that needs a missing free-boundary argument; the gap is probably fixable, so let it go to referees. read the letter →

arxiv 2507.14112 v1 pith:2SG4DPIT submitted 2025-07-18 math.AP math.DG

Matteo Novaga , Emanuele Paolini , Vincenzo Maria Tortorelli This is my paper · ORCID

classification math.APmath.DG MSC 49Q2049Q0549Q10

keywords isoperimetricpartitionslenspartitionSimonsconesingularminimalblow-downconcentrationcompactnessperimeterdefecthalf-space

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that the standard “lens” does not describe all perimeter-minimizing three-way partitions of Euclidean space once the dimension reaches 8. It constructs an isoperimetric 3-partition of $\mathbb{R}^8$ with one region of volume 1 and two regions of infinite volume, and shows this partition is asymptotic to a singular minimal cone rather than to a pair of half-spaces meeting in a hyperplane. This matters because for dimensions up to 7 all such triple partitions are known to be lenses; the result locates the first dimension where the lens model breaks, in parallel with the known appearance of singular minimal cones in dimension 8. The proof works by comparing two numbers: the “defect” needed to turn a hyperplane into a lens is strictly larger than the defect needed to turn the Simons’ cone into a “barrel” partition, and this gap forces at least one limit partition to be non-lens.

What carries the argument

The load-bearing device is the defect functional of Definition 5.1: for a partition $E$ with unique blow-down $E_\infty$, the defect $\Delta^{\Pi}_{E}$ is the limiting excess perimeter per unit of $|E_1|^{(d-1)/d}$ inside the cone $\Pi$. The proof’s engine is the strict numerical comparison in Lemma 5.6: the lens partition’s defect is $\approx 7.29$ while the barrel partition—the Simons’ cone with a $B^4\times B^4$ block inserted—has defect $\approx 7.10$. A compactness and closure theorem (Theorem 3.6) transfers isoperimetric minimality to limits in half-spaces, a monotonicity formula gives blow-downs in cones (Corollary 4.4), and Proposition 5.10 asserts that the only possible half-space limits contribute at least the lens defect; the gap $\Delta_Q < \Delta_L$ then makes the contradiction go through.

What would settle it

Recompute the closed-form defects in Lemma 5.6: the argument requires the lens defect $\Delta_L \approx 7.29$ to be strictly larger than the barrel defect $\Delta_Q \approx 7.10$, and if the exact values instead satisfy $\Delta_L \le \Delta_Q$ the contradiction in Theorem 5.12 loses its force. A separate direct check would be to produce a half-space minimal interface meeting the boundary at a non-right angle, which would falsify the reflection step of Proposition 5.10.

Watch

Extended reading notes

Core claim

The central result is Theorem 5.12: there exists an isoperimetric 3-partition $E=(E_1,E_2,E_3)$ of $\mathbb{R}^8$ with $m(E)=(1,+\infty,+\infty)$ which is not a lens partition. Its blow-down partition $E_\infty=(\emptyset,E_1^\infty,E_2^\infty)$ is a singular minimal cone. The construction takes, for large radii $R_n$, the isoperimetric partition in the ball $\overline{B}_{R_n}$ that matches the Simons’ cone outside the ball and has $|E_1|=1$; by concentration compactness the rescaled partitions split into limits, each isoperimetric in a half-space or in $\mathbb{R}^8$. If every limit were a lens, summing their defects would give a lower bound $\Delta_L\approx 7.29$, but the barrel partition is a competitor whose defect is $\approx 7.10$, a contradiction. Hence one limit is not a lens; by known uniqueness results its blow-down must be a singular cone.

Load-bearing premise

The proof of Proposition 5.10 assumes that a partition that is locally perimeter-minimal on each side of a boundary plane, with a regular interface, can be reflected across that plane to give a globally stationary interface; the reflection is valid only if the interface meets the plane orthogonally, and that orthogonality is not proved.

Editorial extensions

If this is right

For $N=3$, the lens is not the only isoperimetric partition in $\mathbb{R}^8$: uniqueness of isoperimetric triple partitions fails from dimension 8 onward.
Because the blow-down is a singular minimal cone, the example ties the failure of lens uniqueness to the same dimension threshold where singular minimal cones such as the Simons’ cone first appear for locally minimal sets.
The construction shows that half-space isoperimetric partitions can be non-standard when $d\geq 4$, a phenomenon isolated in Proposition 5.10 and the surrounding discussion.
The method reduces the existence question to comparing two explicit constants, the barrel and lens defects, so the same scheme could in principle be repeated with other candidate cones once the half-space defect bound is available.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the reflection step in Proposition 5.10 can be repaired or replaced, the same defect comparison should produce non-lens isoperimetric 3-partitions in every dimension $d\geq 8$ where an analogous singular cone and barrel can be built.
A natural next step is to identify the non-lens limit partition explicitly; the authors’ numerics suggest it is the constant-mean-curvature deformation of the barrel shown in Figure 1, so proving that profile is isoperimetric would make the construction explicit rather than a limit argument.
The open question whether the blow-down is exactly the Simons’ cone could be settled by computing the density of the constructed blow-down and comparing it with the known density of the Simons’ cone.
One might expect that, for all sufficiently large radii, the finite-ball minimizers themselves are already non-lens; the paper only proves that at least one limit is non-lens, and a uniform version of the defect gap would be needed to promote the conclusion to the finite-radius minimizers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Read the thing. The headline result is what it says: a non-standard isoperimetric 3-partition in R^8, one bounded region, two unbounded, asymptotic to a singular cone. That resolves a conjecture that [7] and [17] had left open, and it's the first example for N=3. The construction is very much in the spirit of the Simons cone: compare the 'defects' of the lens vs. the barrel, then use concentration compactness on the approximating balls. The defect arithmetic in Lemma 5.6 checks out (7.29 vs 7.10), and the idea of forcing a non-lens limit by ruling out all-lens limits via a defect lower bound is genuinely new and quite elegant. The paper is honest about its limits—it doesn't prove the limit is the Simons cone, and it flags the independent Bronsard et al. result.

The soft spot is where the stress test points: Proposition 5.10. The proof reflects the half-space interface and asserts that the reflected varifold is stationary in R^d\B_R. That doesn't follow from the two displayed local minimalities. You need the interface to meet ∂Π orthogonally, i.e., the free-boundary condition. The paper says ∂E is regular on ∂Π and then jumps to stationarity. Regularity alone doesn't imply the crease disappears. The isoperimetric property in the closed half-space should force the orthogonality by a standard first-variation argument, but it's not supplied. Since Theorem 4.1, the asymptotic expansion, and the bound Δ≥Δ_L all ride on this, it's a real gap in the written proof—not a cosmetic one.

I don't think it's fatal. The missing argument is likely a page of standard variations and a citation to Allard or Hardt-Simon. But the paper as submitted is not complete. Lemma 5.7 is also hard to unwind; the ε bookkeeping looks plausible but I wouldn't want to certify it without doggedly tracing the estimates.

Who is this for? Geometric measure theorists working on isoperimetric clusters, and anyone interested in the Bernstein/Simons analogy for partitions. It deserves the full referee treatment, but the referee should demand the Prop 5.10 fix before publication.

Recommendation: send to peer review, conditionally accept if the free-boundary stationarity is proven.

Best, [You]

Referee Report

2 major / 4 minor

Summary. The paper proves existence of an isoperimetric 3-partition of R^8 with one bounded region of volume 1 and two unbounded regions, which is not a lens partition and whose blow-down is a singular minimal cone. The strategy is to compare the "defect" of a lens partition (≈7.29) with that of a barrel partition built from the Simons cone (≈7.10), then take limits of isoperimetric partitions in large balls with Simons-cone boundary data. Concentration compactness produces limit partitions in R^8 or in half-spaces; if all such limits were lenses or lens-like half-space partitions, the total defect would be bounded below by that of a lens, contradicting the upper bound from the barrel partition. The proof relies on the authors' earlier framework for compactness and closure of isoperimetric partitions, and on several new results for half-space partitions.

Significance. If the main theorem is correct, it resolves a conjecture raised in the authors' earlier work and in [7], and provides the first non-standard isoperimetric 3-partition in Euclidean space, in analogy with the Simons-cone counterexamples for minimal sets. The explicit defect computations in Lemma 5.6 are a concrete strength: they are analytic, checkable, and give the numerical ordering that drives the argument. The paper also extends concentration-compactness and blow-down tools to partitions with half-space limits, which is of independent interest. However, the proof as written contains a load-bearing gap in the free-boundary stationarity assertion of Proposition 5.10.

major comments (2)

[Proposition 5.10, paragraph 2] The assertion that the reflected set E = (F2∩Π) ∪ σ(F2∩Π) has stationary boundary varifold v(∂E\BR,1) in Rd\BR is not justified. Local minimality of F in Π controls variations supported in Π, and local minimality of the reflected copy controls variations in the complementary half-space, but neither controls variations supported in a ball centered on ∂Π that cross the boundary. Stationarity across ∂Π is a separate first-variation statement equivalent to the free-boundary condition that the interface meets ∂Π orthogonally; the proof never derives this condition from the J-isoperimetric property in Definition 2.2, and the regularity of ∂E on ∂Π does not by itself force it. Consequently the hypotheses of Theorem 4.1 are not verified, so the asymptotic expansion (49), the convergence (76), and the bound ∆^Π_F ≥ ∆L in (77) are unsupported. Since (77) is used to establish (81) in Theorem 5.12, the only route to the contradiction ∆L ≤ ∆Q, this gap is load-bearing for the main result.
[Theorem 5.8] The proof applies the strong maximum principle of [21, Corollary 1] to ∂E∞, where E∞ is a locally minimal cone that may be singular at the origin (in R^8, Simons-type cones are singular). The quoted result is formulated for smooth area-minimizing hypersurfaces, and the manuscript does not justify its applicability at singular points of ∂E∞. This matters because Theorem 5.9 and, through it, the case analysis in Theorem 5.12 use Theorem 5.8 to identify blow-downs as half-planes. The authors should either state a maximum principle valid for minimal cones with singularities, or give a separate argument (for example, using tangent cones at regular points) that ∂E∞ must coincide with ∂Π.

minor comments (4)

[Abstract and title] The abstract contains a grammatical error: "a isoperimetric" should be "an isoperimetric"; the title also has a spacing artifact in "P AR TITION".
[Definition 5.4] In the definition of the Simons' cone partition, the set S3 is written as Rd \ S2, but since the ambient space is R^8 the notation should be R^8 \ S2 for consistency.
[Lemma 5.6] The displayed formula for ∆L in the proof contains the garbled expression "4 8 s 4π3...", which appears to be a typographical artifact of the radical notation; the formula should be typeset as a single clean radical expression.
[Theorem 5.12] In the statement of the theorem, the notation "E∞ = (∅, E1∞, E2∞)" is inconsistent with the convention used elsewhere for partitions, where regions are indexed by subscripts (E1, E2, E3); this should be clarified or aligned with the earlier notation.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main contradiction is driven by analytically computed defect inequalities and external uniqueness/blow-down results, not by assumptions equivalent to the conclusion.

full rationale

The derivation chain in Theorem 5.12 is not circular. The proof assumes toward contradiction that every blow-up limit F_i is a lens or satisfies the half-space defect bound Δ^{Π_i}_{F_i} ≥ Δ_L (Eq. 81), then uses Lemma 5.7 to obtain limsup Δ_{E_n} ≥ Δ_L and the barrel competitor to obtain Δ_{E_n} ≤ Δ_Q, contradicting the strict analytic inequality Δ_L ≈ 7.29 > Δ_Q ≈ 7.10 computed directly in Lemma 5.6 from exact volumes and perimeters. No fitted parameter is renamed as a prediction: the defect values are computed analytically, not calibrated, and the contradiction is genuine. The compactness and closure machinery (Theorem 3.1, Theorem 3.6) is either proved in the paper or adapted with proofs from the authors' earlier published work; those cited results are parameter-free and do not assume the target partition. External inputs [7, 12, 10, 21] are uniqueness, flatness-expansion, and boundary-regularity theorems independent of the present construction. The one genuinely questionable inference is in Proposition 5.10, where the reflected set E = (F2∩Π) ∪ σ(F2∩Π) is claimed to have stationary varifold boundary across ∂Π from local minimality in each half-space plus boundary regularity; the required free-boundary orthogonality is not proved, and the subsequent use of Theorem 4.1, expansion (49), convergence (76), and the inequality Δ^{Π}_F ≥ Δ_L depends on it. That is a potential correctness gap, not circularity: it does not assume the conclusion Δ^{Π}_F ≥ Δ_L, does not rename a fitted quantity, and is not a self-citation chain. Therefore no circular step is identified.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central proof introduces no fitted parameters and no new objects beyond the known lens, Simons cone, and barrel partitions. The numerical defect values are analytic computations from explicit formulas, not fitted. The main external inputs are established theorems in geometric measure theory and prior classification results, the most load-bearing being [7] and [17].

assumptions (6)

standard math Standard compactness and semicontinuity results for Caccioppoli partitions, including the concentration compactness theorem (Theorem A.3).
Used in Section 3 and Theorem 5.12 to extract limit partitions from the approximating sequence.
domain assumption Existence and closure of isoperimetric partitions with conical boundary data, built on the framework of [17].
Theorem 3.6 and the volume-fixing Lemma 3.4 justify the competitor constructions in Lemma 5.7 and the main proof.
standard math Uniqueness of isoperimetric 3-partitions in R^d for d ≤ 7, proved by Bronsard and Novack in [7].
Used in Theorem 5.12 to assert that any whole-space limit with planar blow-down must be a lens partition.
standard math Strong maximum principle for minimal hypersurfaces from [21].
Used in Theorem 5.8 to force the blow-down of a locally minimal set contained in a half-space to coincide with the boundary hyperplane.
standard math Boundary regularity and C^{1,α} regularity for locally minimizing currents from [10] and [3].
Used in Theorem 5.9 to conclude that the blow-down interface over a half-space is a half-hyperplane.
standard math Asymptotic expansion of planelike stationary varifolds from [12] and [2].
Used in Proposition 5.10 to show asymptotic flatness of the reflected set E.

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Pith. "Pith review of Existence of a non-standard isoperimetric triple partition." pith.science (2026). https://pith.science/paper/2SG4DPIT

@misc{pith2026250714112,
  author       = {Pith},
  title        = {Pith review of: Existence of a non-standard isoperimetric triple partition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2SG4DPIT}},
  note         = {Machine review of arXiv:2507.14112}
}

abstract

We show existence of a isoperimetric $3$-partition of $\mathbb R^8$, with one set of finite volume and two of infinite volume, which is asymptotic to a singular minimal cone.

Figures

Figures reproduced from arXiv: 2507.14112 by the authors.

**Figure 1.** Numerically computed picture of the conjectured isoperimetric 3-partition of R 8 . The actual partition in R 8 would be the triple (r(E1), r(E2), r(E3)) where E1, E2, E3 ⊆ R 2 are depicted in the figure, and r(E) = (x, y) ∈ R 4 × R 4 : (|x| , |y|) ∈ E [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** A reference picture for the proof of Theorem 3.6. Our aim is to prove that the inequality (15) is valid up to an error of order ε. This will be obtained by building a cluster Eˆ n competing to the minimality of En for some large n. First a cluster E˜ n is obtained by patching all the given clusters Gi into large holes Ωi n = BR−ρ(x i n ) \ BRn cut from En. The radius R > 0 is large enough to contain all the variatio… view at source ↗

**Figure 3.** A reference picture for the proof of Lemma 5.7. F i ∞ = (∅,(F i ∞)2,(F i ∞)3), Ti = (∅, Ti 2 , Ti 3 ), Hi = (∅, Hi 2 , Hi 3 ), with ∂Hi = ∅ or ∂Hi is a hyperplane passing through the origin, so that F i r L 1 loc −→ F i (52) ∞ as r → +∞ S − x i n L 1 loc −→ T i (53) as n → +∞ (F i ∞ + y i ) ∩ Π i = (Hi + y i ) ∩ Π i (54) (F i ∞ + y i ) \ Π i = T i \ Π i (55) , for some y i ∈ R d . Suppose also (56) [PITH_FULL_IMAGE… view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Anisotropic isoperimetric double tilings of the plane
math.AP 2026-07 accept novelty 6.5 of 10

Under L1 perimeter the isoperimetric double-tiling profile is 2√x + 2√(1-x), uniquely realized by the Pythagorean tiling of two axis-aligned squares except at equal area.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [7]

Lia Bronsard and Michael Novack, An infinite double bubble theorem, 2024, ArXiv:2401.08063

work page Pith review arXiv 2024

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Matteo Novaga, Emanuele Paolini, and Vincenzo Maria Tortorelli, Locally isoperimetric par- titions, Trans. Amer. Math. Soc. 378 (2025), no. 4, 2517–2548

work page 2025

[3] [1]

Stan Alama, Lia Bronsard, and Silas Vriend, The standard lens in R2 uniquely minimizes relative perimeter, Trans. Amer. Math. Soc. 12 (2025), 516–535

work page 2025

[4] [2]

Allard, On the first variation of a varifold , Ann

W. Allard, On the first variation of a varifold , Ann. of Math. 95 (1972), 417–491

work page 1972

[5] [3]

Allard, On the first variation of a varifold: Boundary behavior , Annals of Math- ematics 101 (1975), no

William K. Allard, On the first variation of a varifold: Boundary behavior , Annals of Math- ematics 101 (1975), no. 3, 418–446

work page 1975

[6] [4]

Bombieri, E

E. Bombieri, E. De Giorgi, and E. Giusti, Minimal cones and the Bernstein problem , Inven- tiones math. 7 (1969), 243–268

work page 1969

[7] [5]

Brakke, Minimal cones on hypercubes , J

Kenneth A. Brakke, Minimal cones on hypercubes , J. Geom. Anal. 1 (1991), no. 4, 329–338

work page 1991

[8] [6]

Lia Bronsard, Robin Neumayer, Michael Novack, and Anna Skorobogatova, On the non- uniqueness of locally minimizing clusters via singular cones , 2025

work page 2025