The Makai inequality in higher dimensions: qualitative and quantitative aspects

Abstract: In this paper, given a convex, bounded, open set $Ω\subset \mathbb{R}<sup>n$ 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.

• The paper proves the sharp upper bound for the geometric functional F(Ω)=T(Ω)P²(Ω)/|Ω|³, confirming a longstanding conjecture in convex domains.
• It demonstrates that no single convex domain attains the supremum, as maximizing sequences converge to flattening tangential bodies with vanishing thickness.
• A quantitative inequality is derived linking the optimality deficit to intrinsic geometric parameters by introducing explicit remainder terms like γ(Ω).

The Makai Inequality in Higher Dimensions: Qualitative and Quantitative Analysis

Context and Significance

The paper "The Makai inequality in higher dimensions: qualitative and quantitative aspects" (2604.14000) systematically extends and sharpens classical geometric functionals associated with convex domains in $\mathbb{R}^n$. The focus lies on the scaling-invariant quantity

$$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$

where $T(\Omega)$ denotes Laplacian torsional rigidity, $P(\Omega)$ the perimeter, and $|\Omega|$ the Lebesgue measure of a convex open set $\Omega$. This functional encodes fundamental relationships between PDE characteristics and convex geometry and has been previously studied by Pólya and Makai. The contribution not only proves the conjectured sharp upper bound for $\mathcal{F}$ in higher dimensions but also provides a refined quantitative structure for optimizing sequences, enhancing the geometric understanding of extremal and near-extremal domains.

Main Qualitative Results

The principal result confirms a conjecture formulated in [buttazzo2020convex] regarding the sharp upper bound for $\mathcal{F}$ across all dimensions. For every convex, bounded, open set $\Omega \subset \mathbb{R}^n$ ($n \geq 2$), it holds

$$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$0

with the constant proven to be sharp. This value is asymptotically attained by sequences of flattening cones—domains whose thickness vanishes in a controlled manner, precisely characterizing the geometric limit of maximizers.

The analysis also resolves the existence question: the supremum is not achieved by any single set $$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$1, but only in the limit along particular thinning sequences. Formally, if $$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$2 is a maximizing sequence, then the minimal width to diameter ratio $$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$3 satisfies $$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$4. This highlights the impossibility of attaining the optimal constant with a domain of positive thickness.

Quantitative Structure and Characterization

A second distinguished result is a quantitative inequality relating the deficit from optimality to intrinsic geometric parameters. Utilizing the remainder term

$$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$5

where $$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$6 is the inradius, the paper establishes

$$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$7

with explicit $$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$8. This result directly quantifies how far any given convex set is from a flattening tangential body in terms of geometric measurements.

The correlation between $$\mathcal{F}(\Omega) = \frac{T(\Omega) P^2(\Omega)}{|\Omega|^3},$$9, $T(\Omega)$0, and $T(\Omega)$1—three parameters capturing distinct aspects of domain thickness and tangentiality—is rigorously examined. The optimizing sequences for $T(\Omega)$2 are necessarily characterized by $T(\Omega)$3 and $T(\Omega)$4, revealing that the limit domains are flattening tangential bodies, with flattening cones providing an explicit asymptotic realization.

Methodological Framework

The proofs draw from advanced geometric analysis, leveraging coarea formulas, the Brunn–Minkowski theory, and precise estimates for torsional rigidity. Key methodological steps include:

• Concavity arguments: Employing the concavity of $T(\Omega)$5 and $T(\Omega)$6 (inner parallel sets) to bound measures and perimeters of subdomains.
• Sharp integral estimates: Relating torsional rigidity to squared distances from the boundary, via detailed integration by parts and asymptotic expansions.
• Remainder terms: Introduction and calibration of $T(\Omega)$7, $T(\Omega)$8, and $T(\Omega)$9 to obtain both qualitative and quantitative optimality conditions.

Implications and Future Perspectives

The results establish a definitive characterization of extremal behavior for $P(\Omega)$0 in higher dimensions. The sharp constant and quantitative estimates provide a structural foundation for ongoing work on shape optimization, spectral inequalities, and geometric analysis of PDEs.

Contradictory claim: The supremum is never attained within $P(\Omega)$1 for $P(\Omega)$2, contradicting any previous assumption of existence of optimal domains with positive interior.

From a theoretical standpoint, the findings motivate further exploration of tangential and thinning domains, their spectral properties, and connections with other isoperimetric type inequalities and functional optimization under geometric constraints. Applications may be envisioned in elasticity theory, spectral geometry, and probabilistic models (e.g., Brownian motion lifetime asymptotics for thin domains [BFreitas]), as well as in numerical shape optimization for boundary and spectral control.

The methodology, particularly the explicit quantitative deficit bounds, is widely transferable to other variational functionals and may influence new approaches to stability analysis for eigenvalue problems and functional inequalities.

Conclusion

The paper rigorously resolves the optimality and quantitative structure of the Makai inequality for $P(\Omega)$3 in all dimensions, confirming the conjecture regarding the sharp constant and providing a geometric description of asymptotic maximizers. The quantitative deficit bounds introduce a refined geometric lens for analyzing convex domains, with implications for both the theory of PDEs and the geometry of convex sets. The work situates itself as a reference for future research on extremal and near-extremal geometric inequalities for convex domains (2604.14000).