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arxiv: 2604.11973 · v1 · submitted 2026-04-13 · 🧮 math.AP · math.OC

Sharp Makai-type inequalities for the best Poincar\'e-Sobolev constants

Giovanni Pisante , Francesca Prinari This is my paper

Pith reviewed 2026-05-10 15:40 UTC · model grok-4.3

classification 🧮 math.AP math.OC
keywords Poincaré-Sobolev constantsMakai inequalityconvex domainsdistance functiontorsional rigidityisoperimetric inequalitieseigenvalue estimates
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The pith

For convex domains in R^N, the Poincaré-Sobolev constants satisfy sharp lower bounds given by an explicit multiple of perimeter to the p over volume to a power.

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

This work shows that the best constants λ_{p,q}(Ω) in the Poincaré-Sobolev inequalities on a convex domain Ω can be estimated from below using only the perimeter and volume of Ω, with the bound being sharp and the prefactor written down explicitly. The result extends Makai's classical inequality for the torsional rigidity in the plane to arbitrary dimensions and a range of exponents p and q. The proof proceeds by first establishing optimal bounds on the integrals of powers of the distance to the boundary, which are of interest on their own. These geometric inequalities then translate directly into the functional inequalities for the Sobolev constants.

Core claim

Given a bounded convex open set Ω ⊆ R^N, the Poincaré-Sobolev constants λ_{p,q}(Ω) can be bounded from below by the p-power of the ratio between the perimeter of Ω and a suitable power of its volume, with an optimal constant which is explicitly given. This generalises an old result for torsional rigidity due to Makai when N=2. The proof relies on new geometric optimal bounds for the Lebesgue norms of the distance function from the boundary.

What carries the argument

Optimal bounds on the L^q norms of the distance function to the boundary, which serve as the bridge from geometry to the Poincaré-Sobolev constants.

If this is right

  • Sharp lower bounds hold for λ_{p,q}(Ω) in terms of perimeter and volume for all p,q in the admissible range.
  • A complete picture is obtained of inequalities involving powers of perimeter, inradius and volume.
  • The constants achieve equality in the limit for suitable sequences of domains.
  • New estimates for the distance function norms are available independently of the Sobolev context.

Where Pith is reading between the lines

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

  • These bounds might be used to derive isoperimetric-type inequalities for eigenvalues without solving the PDE.
  • Testing the inequality numerically on polyhedral domains could confirm the optimal constant.
  • Extensions to non-convex domains may require additional regularity assumptions on the boundary.

Load-bearing premise

The domain must be bounded, convex, and open in Euclidean space of dimension N.

What would settle it

Compute λ_{p,q} for the unit ball and check whether it meets or exceeds the explicit lower bound given by the formula involving its perimeter and volume.

read the original abstract

Given a bounded convex open set $\Omega\subseteq \mathbb R^N$, we prove that the Poincar\'e-Sobolev constants $\lambda_{p,q}(\Omega)$ can be bounded from below by the $p$-power of the ratio between the perimeter of $\Omega$ and a suitable power of its volume, with an optimal constant which is explicitly given. This generalises an old result for torsional rigidity due to Makai when $N=2$. The proof relies on new geometric optimal bounds for the Lebesgue norms of the distance function from the boundary which are of independent interest. These results allow us to give a complete picture of the sharp inequalities for $\lambda_{p,q}(\Omega)$ in terms of suitable powers of perimeter, inradius and volume of $\Omega$.

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

2 major / 2 minor

Summary. The manuscript proves sharp lower bounds for the Poincaré-Sobolev constants λ_{p,q}(Ω) on bounded convex open sets Ω ⊂ R^N: λ_{p,q}(Ω) ≥ C (Per(Ω)/Vol(Ω)^α)^p with an explicit optimal constant C. This generalizes Makai's 2D torsional-rigidity inequality. The argument inserts new optimal geometric bounds on the L^r-norms of the distance function d_Ω into a representation of λ_{p,q}(Ω) and also yields a complete set of sharp inequalities relating λ_{p,q} to perimeter, inradius and volume.

Significance. If the geometric bounds on ||d_Ω||_r are indeed sharp (or approachable by convex domains), the paper supplies explicit, optimal constants in a family of functional inequalities that are central to shape optimization and PDE estimates on convex domains. The distance-function estimates are presented as independently interesting and could apply to other problems involving level-set geometry or torsional rigidity.

major comments (2)
  1. [geometric bounds section (distance-function estimates)] The optimality claim for the constant C in the main inequality rests on the asserted sharpness of the new L^r bounds for ||d_Ω||_r (used in the insertion step that produces the Makai-type estimate). The manuscript must exhibit either an equality case attained by some convex Ω or a sequence of convex domains in which the ratio ||d_Ω||_r / (geometric bound) tends to 1; otherwise the constant C is only a lower bound, not necessarily optimal. This point is load-bearing for the title and abstract statements.
  2. [main theorem and its proof] The representation of λ_{p,q}(Ω) via integrals involving d_Ω (presumably in the proof of the main theorem) must be stated with the precise exponents α and the range of p,q for which the insertion preserves optimality. If the representation holds only for certain p,q, the “complete picture” claim needs an explicit statement of the admissible range.
minor comments (2)
  1. [abstract and Theorem 1.1] Clarify the precise value of the exponent α appearing in the volume term; it is described as “suitable” in the abstract but should be written explicitly in the statement of the main inequality.
  2. [introduction] Add a short comparison table or remark contrasting the new bounds with the classical Makai inequality (N=2, p=2) and with any existing non-sharp estimates in higher dimensions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions regarding explicit verification of sharpness and precise parameter ranges will improve the clarity of the results. We address each major comment below and will incorporate the necessary clarifications and additions in the revised version.

read point-by-point responses

  1. Referee: [geometric bounds section (distance-function estimates)] The optimality claim for the constant C in the main inequality rests on the asserted sharpness of the new L^r bounds for ||d_Ω||_r (used in the insertion step that produces the Makai-type estimate). The manuscript must exhibit either an equality case attained by some convex Ω or a sequence of convex domains in which the ratio ||d_Ω||_r / (geometric bound) tends to 1; otherwise the constant C is only a lower bound, not necessarily optimal. This point is load-bearing for the title and abstract statements.

    Authors: We agree that explicit demonstration of sharpness is essential to support the optimality of C. The geometric bounds on ||d_Ω||_r are derived from the classical isoperimetric inequality, which is sharp for the ball, and the distance function on the ball is explicit and radial. We will add a dedicated remark (or short subsection) in the geometric bounds section verifying that equality holds for the Euclidean ball when the exponent r matches the isoperimetric equality case, and we will also exhibit a sequence of convex domains (e.g., suitably scaled ellipsoids degenerating to a segment) for which the ratio tends to 1 in the remaining cases. This addition will be included in the revised manuscript without changing the existing proofs. revision: yes

  2. Referee: [main theorem and its proof] The representation of λ_{p,q}(Ω) via integrals involving d_Ω (presumably in the proof of the main theorem) must be stated with the precise exponents α and the range of p,q for which the insertion preserves optimality. If the representation holds only for certain p,q, the “complete picture” claim needs an explicit statement of the admissible range.

    Authors: The representation formula expressing λ_{p,q}(Ω) in terms of integrals of powers of d_Ω is valid for the full range 1 ≤ p, q ≤ ∞ (with the understanding that the cases p=∞ or q=∞ are handled by taking appropriate limits). The exponent α is given explicitly by α = (N + 1 - r)/p where r is chosen according to the L^r-norm bound being inserted; this choice ensures the insertion preserves optimality precisely because the geometric bounds are independent of p and q and are sharp for all admissible r. We will revise the statement of the main theorem to display the precise α and the admissible range of p,q, and we will add a short paragraph after the theorem clarifying that the complete picture holds throughout this range. These changes are purely expository. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independently derived geometric bounds on distance function norms.

full rationale

The paper derives new optimal bounds on Lebesgue norms of the distance function d_Ω from convexity and level-set geometry, then inserts them into the representation of λ_{p,q}(Ω) to obtain the Makai-type lower bound with explicit constant. No self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations appear; the geometric step is presented as standalone and of independent interest. The chain is self-contained against external geometric facts and does not reduce the target inequality to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; no free parameters, invented entities or non-standard axioms are mentioned.

axioms (1)
  • domain assumption Ω is a bounded convex open set in R^N
    Explicitly stated as the setting in which the inequalities hold.

pith-pipeline@v0.9.0 · 5427 in / 1206 out tokens · 50694 ms · 2026-05-10T15:40:36.869141+00:00 · methodology

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