arxiv: 2605.04331 · v1 · submitted 2026-05-05 · 🧮 math.SP · math.AP

Recognition: 3 theorem links

· Lean Theorem

Sharp Dirichlet eigenvalue inequalities on triangles

Phanuel Mariano, Ryoki Endo, Xuefeng Liu

Pith reviewed 2026-05-08 18:30 UTC · model grok-4.3

classification 🧮 math.SP math.AP

keywords Dirichlet eigenvaluestrianglesshape optimizationeigenvalue inequalitiesLaugesen-Siudeja conjectureCheeger inequality

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The pith

The equilateral triangle uniquely minimizes a scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter among all triangles.

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

The paper establishes that the equilateral triangle is the only shape that achieves the smallest value of this combined quantity, which stays the same under scaling. This directly settles an earlier conjecture and supplies the sharpest possible lower bound on the eigenvalue that uses both area and perimeter. The argument relies on a fresh technique for bounding how the functional changes when vertices move, backed by error-controlled numerical checks and separate analysis of triangles that are nearly collapsed into lines.

Core claim

The equilateral triangle uniquely minimizes the scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter. This settles the Laugesen-Siudeja conjecture, yields an optimal two-term lower bound for the first Dirichlet eigenvalue in terms of area and perimeter, and establishes a Cheeger-type inequality with an explicit best constant for triangles.

What carries the argument

A computable lower bound for second-order directional shape derivatives of the functional under vertex perturbations, combined with validated finite-element error estimates and analytic bounds for nearly degenerate triangles.

If this is right

An optimal two-term lower bound holds for the first Dirichlet eigenvalue of any triangle in terms of its area and perimeter.
A Cheeger-type inequality holds for triangles with the explicit best constant.
The same derivative-bound technique applies directly to other Dirichlet eigenvalue inequalities on triangles.

Where Pith is reading between the lines

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

The combination of derivative lower bounds with finite-element validation could be adapted to settle minimization questions for other functionals or for polygons with more sides.
The analytic treatment of nearly degenerate triangles supplies a template that might close gaps in related isoperimetric problems where numerical methods alone lose accuracy.

Load-bearing premise

The computable lower bounds on the second-order shape derivatives, the finite-element error controls, and the analytic estimates for skinny triangles are all accurate enough to rule out any smaller value outside the equilateral case.

What would settle it

Explicit construction or high-precision numerical evaluation of any non-equilateral triangle for which the scale-invariant functional takes a strictly smaller value than it does for the equilateral triangle.

Figures

Figures reproduced from arXiv: 2605.04331 by Phanuel Mariano, Ryoki Endo, Xuefeng Liu.

**Figure 1.** Figure 1: Decomposition of the shape space and contour plots of the two scale-invariant functionals. The proof of Theorem 1.4 is divided into three parts. (i) The region Ωup. First, we prove analytically that ∇Jk(△p0 ) = 0 (k = 1, 2). We then use the computable lower bound (9) to verify ∂ 2Jk ∂x2 > 0 (k = 1, 2), over Ωup. Since Jk is symmetric with respect to reflection across the axis x = 1/2, this convexity impli… view at source ↗

**Figure 2.** Figure 2: Subregions Ωup, Ωmid and Ωdown Step 1 (Region Ωup). (1-1) Using the first-order shape derivative formula, we prove analytically that ∇Jk(△p0 ) = 0 (k = 1, 2), that is, p0 is a stationary point of Jk for k = 1, 2. (1-2) We compute a rigorous lower bound for the second-order derivative ∂ 2 xJk (k = 1, 2) and prove its positivity. The resulting strict convexity in the x-direction implies that any minimizer in… view at source ↗

**Figure 3.** Figure 3: Clusters of eigenvalues Let Ek be the space spanned by the exact eigenfunctions associated with the k-th cluster: (45) Ek = span{un, un+1, . . . , uN }. Similarly, let ui,h ∈ H1 0 (△) be approximations of the exact eigenfunctions ui for i = n, . . . , N, and define the corresponding approximate eigenspace by (46) E h k = span{un,h, un+1,h, . . . , uN,h}. We also define λn,h := min uh∈Eh k ∥∇uh∥ 2 △ ∥uh∥ 2 … view at source ↗

**Figure 4.** Figure 4: Parameter cell Ci,j . For p = (x, y) and Λ ∈ R, we introduce auxiliary functionals B1, B2 corresponding to J1, J2 by B1(p; Λ) := Λ|△p | − π 2 16 |∂△p | 2 |△p| − 7 √ 3π 2 12 (75) , B2(p; Λ) := Λ|△p | − 4π 2 3 + p π √ 3 2 |∂△p | + p 4π|△p| 2 4 |△p| (76) . Note that Jk(△p ) = Bk(p; λ1(△p )) for k = 1, 2. Lemma 4.9. Let Cij be a verification cell in Ωmid with vertices pi,j , pi+1,j , pi,j+1, pi+1,j+1 as … view at source ↗

read the original abstract

We prove sharp Dirichlet eigenvalue inequalities for planar triangles. We settle a conjecture of Laugesen and Siudeja by showing that the equilateral triangle uniquely minimizes a scale-invariant functional of the first Dirichlet eigenvalue, area, and perimeter. Consequences include an optimal two-term lower bound for the first Dirichlet eigenvalue in terms of area and perimeter. We also prove a Cheeger-type inequality with an explicit best constant considered by Parini. To prove these conjectures we propose a new method for proving Dirichlet eigenvalue inequalities on triangles. Our method is based on a new computable lower bound for second-order directional shape derivatives under vertex perturbations. It also uses validated finite-element error estimates and recently developed analytic estimates for eigenvalues of nearly degenerate triangles. The method is not specific to the functionals considered in this paper and it can be used to prove various other eigenvalue inequalities on triangles.

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

They settle the Laugesen-Siudeja conjecture with a hybrid analytic-numerical argument that the equilateral triangle uniquely minimizes the scale-invariant lambda1-area-perimeter functional.

read the letter

The main takeaway is that the equilateral triangle is the unique global minimizer for the scale-invariant functional combining the first Dirichlet eigenvalue, area, and perimeter. This closes the Laugesen-Siudeja conjecture and immediately gives an optimal two-term lower bound on the eigenvalue plus an explicit best constant in the Parini Cheeger-type inequality for triangles. The paper also supplies a reusable method for other eigenvalue inequalities on triangular domains. What is new is the computable lower bound on the second-order directional shape derivative under vertex perturbations, backed by validated finite-element error estimates and paired with analytic controls for nearly degenerate triangles. The hybrid structure lets them treat local minimality away from degeneracy numerically and glue it to the limiting cases analytically. The numerics come with explicit error controls, which is better than many shape-optimization papers that rely on unvalidated computation. The soft spot is exactly where the stress-test note flags: the global uniqueness claim depends on the numerical lower bound being accurate and the sampling dense enough to rule out other local minima in the intermediate aspect-ratio range. If the finite-element error estimate underestimates in some transition zone or if the analytic degenerate bounds do not mesh cleanly, a gap remains. The paper must demonstrate that the covered regions abut without uncovered intervals. This work is aimed at people in spectral geometry who care about explicit constants and minimization problems on polygons. Anyone following eigenvalue bounds or shape derivatives on triangles will find the inequalities and the method directly useful. It deserves a serious referee because it resolves a concrete conjecture with a fresh technique, even though the numerical rigor will require close checking.

Referee Report

2 major / 2 minor

Summary. The paper proves sharp Dirichlet eigenvalue inequalities for planar triangles. It settles the Laugesen-Siudeja conjecture by establishing that the equilateral triangle is the unique minimizer of a scale-invariant functional combining the first Dirichlet eigenvalue, area, and perimeter. Consequences include an optimal two-term lower bound for the eigenvalue in terms of area and perimeter, and a Cheeger-type inequality with an explicit best constant. The proof introduces a new method relying on a computable lower bound for second-order directional shape derivatives under vertex perturbations (obtained via validated finite-element computations), combined with analytic estimates for eigenvalues of nearly degenerate triangles.

Significance. If the central claims hold, the work resolves an open conjecture in spectral geometry and supplies explicit sharp constants for eigenvalue inequalities on triangles, which are of independent interest. The proposed method—combining rigorous numerical validation of shape derivatives with analytic estimates for degenerate cases—is general and could apply to other eigenvalue problems on polygons. The use of validated FEM error estimates and explicit handling of the degenerate limit are strengths that enhance rigor.

major comments (2)

[Section describing the computable lower bound and FEM validation (likely §4 or §5)] The global uniqueness claim rests on the numerical lower bound for the second-order directional derivative being strictly positive for all non-equilateral triangles (to establish local minimality) and on the analytic estimates covering the degenerate regime. The manuscript must specify the precise ranges of aspect ratios or perturbation parameters where the FEM-validated bound applies, the sampling density used to cover the space of triangles, and the explicit error tolerances that guarantee no undetected local minima exist in the transition zone between the numerical and analytic regimes.
[Analytic estimates for nearly degenerate triangles and the global minimization argument] The gluing argument between the numerical lower bound and the analytic estimates for nearly degenerate triangles requires an explicit overlap or matching region where both bounds are valid and the minimum of the two is still positive away from the equilateral case. Without a concrete statement of the transition threshold (e.g., a specific bound on the smallest angle or side-length ratio) and verification that the functional value in that region exceeds the equilateral value, the global minimization is not fully closed.

minor comments (2)

[Introduction and statement of main results] Notation for the scale-invariant functional should be introduced once and used consistently; currently the combination of λ1, area, and perimeter appears in several equivalent forms without a single defining equation.
[Numerical results section] The figures showing eigenvalue contours or derivative surfaces would benefit from explicit axis labels indicating the range of vertex perturbations sampled and a statement of the validated error bound in the caption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comments that will improve the clarity and rigor of the presentation. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Section describing the computable lower bound and FEM validation (likely §4 or §5)] The global uniqueness claim rests on the numerical lower bound for the second-order directional derivative being strictly positive for all non-equilateral triangles (to establish local minimality) and on the analytic estimates covering the degenerate regime. The manuscript must specify the precise ranges of aspect ratios or perturbation parameters where the FEM-validated bound applies, the sampling density used to cover the space of triangles, and the explicit error tolerances that guarantee no undetected local minima exist in the transition zone between the numerical and analytic regimes.

Authors: We agree that these implementation details should be stated explicitly to make the coverage of the numerical validation fully transparent. The current manuscript describes the validated FEM approach and the resulting lower bound but does not consolidate the precise ranges, sampling grid, and error tolerances in one location. In the revision we will add a dedicated paragraph (or subsection) that records: the exact interval of aspect ratios (or normalized vertex perturbations) to which the FEM bound applies, the sampling density and grid used to check positivity over the space of triangles, and the validated error tolerance that ensures the computed lower bound remains strictly positive. This addition will confirm there are no gaps that could hide local minima in the transition region. revision: yes
Referee: [Analytic estimates for nearly degenerate triangles and the global minimization argument] The gluing argument between the numerical lower bound and the analytic estimates for nearly degenerate triangles requires an explicit overlap or matching region where both bounds are valid and the minimum of the two is still positive away from the equilateral case. Without a concrete statement of the transition threshold (e.g., a specific bound on the smallest angle or side-length ratio) and verification that the functional value in that region exceeds the equilateral value, the global minimization is not fully closed.

Authors: We concur that an explicit transition threshold and verification in the overlap region are necessary to close the global argument rigorously. The manuscript currently invokes the analytic estimates for sufficiently degenerate triangles and the numerical bound elsewhere, but the precise matching region and a direct check that the functional exceeds the equilateral value throughout that region are not stated. In the revised version we will introduce a concrete threshold (in terms of the smallest angle or side-length ratio) that defines the overlap, and we will supply a short analytic or computational verification that the combined lower bound remains positive and strictly larger than the equilateral value in this zone. This will complete the gluing step without altering the underlying results. revision: yes

Circularity Check

0 steps flagged

No circularity: proof uses independent validated FEM bounds and external analytic estimates for global uniqueness.

full rationale

The central claim settles an external conjecture (Laugesen-Siudeja) via a new method: a computable lower bound on second-order directional shape derivatives under vertex perturbations, combined with validated finite-element error estimates and analytic estimates for nearly degenerate triangles. These components are derived from first principles or independent numerical validation rather than fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations. The derivation chain does not reduce any key inequality or uniqueness statement to its own inputs by construction; the numerical bounds are cross-validated against error estimates and glued to analytic limits without circular renaming or ansatz smuggling. This is a standard non-circular case relying on external benchmarks and rigorous computation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical properties of Dirichlet eigenvalues and the validity of the new lower-bound method plus error estimates; no free parameters or invented entities are indicated in the abstract.

axioms (2)

standard math Standard properties of Dirichlet eigenvalues and their dependence on domain shape
Invoked for the functional, inequalities, and shape derivatives throughout.
domain assumption Accuracy of validated finite-element error estimates and analytic estimates for degenerate triangles
Load-bearing for the numerical and limiting-case parts of the proof.

pith-pipeline@v0.9.0 · 5434 in / 1401 out tokens · 61548 ms · 2026-05-08T18:30:36.075549+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.

Cost.FunctionalEquation (Jcost / washburn_uniqueness_aczel) washburn_uniqueness_aczel unclear
F(△) := λ₁(△)|△| − (π²/16)·|∂△|²/|△| ... minimized uniquely by the equilateral triangle, and the minimum value is 7√3π²/12.

Reference graph

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