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arxiv: 2604.14000 · v1 · submitted 2026-04-15 · 🧮 math.AP · math.SP

The Makai inequality in higher dimensions: qualitative and quantitative aspects

Vincenzo Amato , Nunzia Gavitone , Rossano Sannipoli This is my paper

Pith reviewed 2026-05-10 12:17 UTC · model grok-4.3

classification 🧮 math.AP math.SP
keywords Makai inequalitytorsional rigidityconvex domainshigher dimensionsquantitative estimatesperimeterLaplacianshape optimization
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The pith

Convex domains in any dimension satisfy a sharp inequality that bounds Laplacian torsional rigidity by perimeter and measure.

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

The paper proves a sharp inequality for the Laplacian torsional rigidity of any convex bounded open set in R^n, expressed using the domain's perimeter and Lebesgue measure. This extends the two-dimensional Makai inequality to arbitrary dimensions and supplies quantitative estimates on how optimizing sequences behave geometrically. A reader would care because the result ties a solution of a PDE (the torsion problem) directly to basic geometric quantities, yielding explicit control that was previously known only in the plane. The estimates further describe the thickness and structure of sequences that nearly achieve equality.

Core claim

Given a convex, bounded, open set Ω ⊂ R^n, a sharp inequality holds that relates the Laplacian torsional rigidity of Ω to its perimeter and its measure; the inequality is the direct higher-dimensional generalization of Makai's planar result, and the paper supplies quantitative estimates that describe the geometric structure and thickness of sequences approaching the equality case.

What carries the argument

The Makai inequality in higher dimensions, which furnishes an explicit upper bound on torsional rigidity in terms of perimeter and volume that becomes sharp precisely when the domain is convex.

If this is right

  • The same sharp constant works uniformly across all dimensions.
  • Sequences of convex domains that nearly saturate the inequality must become increasingly close to a specific geometric configuration in the limit.
  • The quantitative estimates give explicit control on how the inradius or minimal width of nearly optimal domains behaves.
  • The inequality supplies a new isoperimetric-type constraint usable in shape-optimization problems governed by the torsion equation.

Where Pith is reading between the lines

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

  • The same convexity assumption might allow analogous sharp bounds for other spectral quantities such as the first Dirichlet eigenvalue.
  • The thickness estimates on optimizing sequences could be turned into a stability result that quantifies the distance to the equality case in terms of the deficit.
  • Removing convexity would likely require a weaker constant or an additional term that penalizes non-convexity.

Load-bearing premise

The domain must be convex, because the proof and the sharpness of the constant both rely on convexity.

What would settle it

A single convex bounded open set in R^3 for which the ratio of torsional rigidity to the product of perimeter and a power of the measure exceeds the constant derived in the paper.

read the original abstract

In this paper, given a convex, bounded, open set $\Omega \subset \mathbb{R}^n$ we prove a sharp inequality involving the Laplacian torsional rigidity and both the perimeter and the measure of the domain. Our result generalizes to arbitrary dimensions the inequality established by Makai in the plane which, as conjectured in arXiv:2007.02549. Furthermore, we establish quantitative estimates that provide key insights into the geometric structure and the thickness of the underlying optimizing sequences.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript proves a sharp inequality relating the torsional rigidity T(Ω), perimeter P(Ω), and measure |Ω| for convex bounded open sets Ω ⊂ R^n. The result generalizes Makai's 2D inequality to arbitrary dimensions and supplies quantitative estimates on the thickness of optimizing sequences of domains.

Significance. If the central proof is correct, the work extends a classical isoperimetric-type inequality to higher dimensions while adding quantitative geometric information on near-extremal domains. The convexity assumption is essential for sharpness, and the thickness estimates provide concrete control that could support stability or convergence arguments in related problems.

minor comments (2)
  1. The citation to the conjecture in arXiv:2007.02549 appears only in the abstract; a brief discussion of the 2D case and the status of the conjecture should be added to the introduction for context.
  2. Notation for the torsional rigidity is introduced as T(Ω) but should be restated explicitly when the inequality is first stated in the main theorem.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation is a self-contained mathematical proof

full rationale

The paper establishes a sharp inequality relating torsional rigidity T(Ω), perimeter P(Ω), and measure |Ω| for convex bounded open sets Ω in R^n by direct proof, generalizing Makai's planar result. The argument relies on the convexity hypothesis to achieve sharpness and on quantitative thickness estimates for optimizing sequences to control limiting behavior. No step reduces a claimed prediction or result to a fitted parameter, self-definition, or load-bearing self-citation by construction; the reference to the conjecture in arXiv:2007.02549 functions only as external motivation. The derivation chain therefore remains independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger is based on standard assumptions for convex domains and torsional rigidity; no free parameters or invented entities are indicated.

axioms (1)
  • standard math Standard properties of convex, bounded, open sets in Euclidean space R^n and definitions of Laplacian torsional rigidity, perimeter, and measure.
    Invoked implicitly for the domain Ω and the inequality statement.

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discussion (0)

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Reference graph

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20 extracted references · 20 canonical work pages

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