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On the optimal sets in Pólya and Makai type inequalities

This paper establishes new quantitative bounds for shape functionals involving torsional rigidity and the first Dirichlet-Laplacian eigenvalue on bounded, open, and convex sets in $\mathbb{R}^n$ , providing key insights into the behavior and structural properties, such as thickness, of their optimizing sequences.

Original authors: Vincenzo Amato, Nunzia Gavitone, Rossano Sannipoli

Published 2026-02-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Vincenzo Amato, Nunzia Gavitone, Rossano Sannipoli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are an architect trying to design the most efficient building possible, but you have a strict rule: the building must be made of a single, solid, convex shape (no holes, no dents). You have two main goals:

Torsional Rigidity: How much can the building twist before it breaks? (You want this to be as high as possible).
The First Eigenvalue: How "loud" is the building's natural hum? (You want this to be as low as possible).

Mathematicians have known for a long time that if you fix the volume (size) of your building, the perfect shape for these goals is a sphere (or a ball in 3D). This is the "Goldilocks" shape.

But here is the twist in this paper:
The authors aren't looking at fixed volumes. They are looking at a different set of rules where the "perfect" shape doesn't actually exist. Instead, the best you can do is to get closer and closer to a specific limit by making your shape thinner and thinner, like a piece of paper or a flat pancake.

Think of it like trying to find the "flattest" pancake in the universe. You can keep pressing it down, getting flatter and flatter, but you never actually reach a state of "zero thickness" because that would mean the pancake disappears. The paper asks: "As we press these shapes flatter, exactly how fast do our efficiency scores improve?"

The Two "Rulers" for Flatness

To measure how close a shape is to being a perfect, infinitely thin slab, the authors invent two different rulers:

The "Width-to-Width" Ruler ( $\alpha$ ): This measures the ratio of the shape's thinnest point (width) to its longest point (diameter). If you have a long, skinny noodle, this number is tiny.
The "Perimeter-to-Volume" Ruler ( $\beta$ ): This measures how much "crust" (perimeter) your shape has compared to its "filling" (volume). If you have a very flat pancake, you have a huge amount of crust relative to the tiny amount of filling inside. This number gets large as the shape flattens.

The Big Discovery

The paper proves that these two rulers are not the same thing. You can have a shape that looks flat on one ruler but not the other.

The Analogy: Imagine a collapsing pyramid (like a tent falling down). It gets very thin (good on Ruler 1), but its perimeter doesn't change in the same way as a flat pancake (bad on Ruler 2).
The authors show that for the specific math problems they are solving, the "Perimeter-to-Volume" ruler ( $\beta$ ) is the true key. If you want to know how close a shape is to the theoretical limit, you must look at how much "crust" it has relative to its "filling."

The Main Results (The "Recipes")

The authors create new mathematical formulas (inequalities) that act like recipes. These recipes tell you:

If your shape is almost a flat slab (high $\beta$ ), then your efficiency score (torsion or eigenvalue) will be very close to the theoretical limit.
If your efficiency score is very close to the limit, then your shape must be a flat slab.

They prove that the relationship between "flatness" and "efficiency" is not random; it follows a strict power law. For example, if you make your shape twice as flat (in terms of the $\beta$ ruler), your efficiency score improves by a specific, predictable amount (like squaring or cubing the flatness).

Why Does This Matter?

In the real world, we often deal with materials that are thin, like sheets of metal, membranes, or biological cells.

Engineers can use these formulas to predict how a thin material will behave without having to build a perfect model.
Mathematicians now have a precise map of the "landscape" of shapes. They know that if you are trying to optimize a shape, you don't need to look at every weird shape in the universe; you just need to look at shapes that are getting flatter and flatter, and they can predict exactly how the math will behave as you get there.

The "Counter-Intuitive" Part

The paper also highlights a fascinating quirk: You cannot always reverse the logic.
If you have a shape that is very flat (low width), it doesn't guarantee that it has the specific "crust-to-filling" ratio needed to hit the mathematical limit. It's like saying, "Just because a car is going fast doesn't mean it's on the highway; it could be on a racetrack." The authors show that some shapes (like collapsing pyramids) get thin but fail to hit the specific efficiency targets that flat pancakes do.

Summary

This paper is a guidebook for the "Flatness" of shapes. It tells us that for certain physical and mathematical problems, the ultimate limit isn't a sphere, but an infinitely thin slab. The authors provide the exact mathematical "speedometer" to tell us how close any given shape is to that limit, proving that the relationship between a shape's geometry and its physical properties is precise, predictable, and governed by the ratio of its surface area to its volume.

1. Problem Statement

The paper investigates shape optimization problems involving specific functionals that combine the torsional rigidity ( $T(\Omega)$ ) and the first Dirichlet-Laplacian eigenvalue ( $\lambda(\Omega)$ ) for bounded, open, and convex sets $\Omega \subset \mathbb{R}^n$ ( $n \ge 2$ ).

Unlike classical isoperimetric inequalities (e.g., Faber-Krahn or de Saint-Venant) where the optimal shape is a ball, the functionals studied in this paper do not admit an optimal set within the class of convex sets. Instead, their extrema are approached asymptotically by sequences of "degenerate" or "thinning" domains (e.g., flattening cylinders, collapsing pyramids, or thinning rectangles).

The specific functionals analyzed are:

Pólya Torsion Functional: $J_1(\Omega) = \frac{T(\Omega)P^2(\Omega)}{|\Omega|^3}$
Pólya Eigenvalue Functional: $J_2(\Omega) = \frac{\lambda(\Omega)|\Omega|^2}{P^2(\Omega)}$
Makai Functional: $J_3(\Omega) = \frac{T(\Omega)}{R_\Omega^2 |\Omega|}$
Hersch Functional: $J_4(\Omega) = \lambda(\Omega)R_\Omega^2$

The core problem is to establish quantitative stability estimates for these functionals. Since there is no optimal shape to compare against, the authors aim to characterize the behavior of minimizing sequences using geometric remainder terms that measure how "thin" or "flat" the domain is.

2. Methodology

The authors employ a combination of variational methods, geometric analysis, and integral inequalities.

Remainder Terms: Two key geometric quantities are introduced to measure the deviation from the optimal limit:
- $\alpha(\Omega) = \frac{w_\Omega}{\text{diam}(\Omega)}$ : The ratio of minimal width to diameter (measures "thinning").
- $\beta(\Omega) = \frac{P(\Omega)R_\Omega}{|\Omega|} - 1$ : A normalized ratio involving perimeter, inradius, and volume.
- Key Insight: The paper proves that $\beta(\Omega) \ge C_0(n)\alpha(\Omega)$ , but the reverse is not true. Thus, $\beta$ provides a more refined characterization of the minimizing sequences for these specific functionals.
Test Functions and Coarea Formula:
- The proofs rely heavily on constructing specific test functions based on the distance to the boundary, $d(x, \partial\Omega)$ .
- The Coarea Formula is used to transform integrals over the domain $\Omega$ into integrals over the level sets of the distance function (inner parallel sets $\Omega_t$ ).
- The authors utilize the concavity of the distance function and properties of the perimeter of inner parallel sets $P(t)$ .
Key Technical Lemmas:
- Lemma 3.1: Establishes bounds on the measure $\mu(t)$ and perimeter $P(t)$ of the inner parallel set at the specific level $t = |\Omega|/P(\Omega)$ . This level is chosen because it relates directly to the remainder term $\beta(\Omega)$ .
- Proposition 2.4: Provides lower and upper bounds for torsion $T(\Omega)$ using integrals of the distance function and its level sets (Web functions).
Sharpness Analysis: The authors construct specific sequences of sets (e.g., thinning parallelepipeds $\Omega_a = [0,1]^{n-1} \times [0,a]$ and collapsing pyramids) to prove that the exponents in their inequalities are optimal and to demonstrate cases where reverse inequalities fail.

3. Key Contributions and Results

A. Characterization of Minimizing Sequences

The paper proves that for the Pólya torsion, Pólya eigenvalue, and Makai functionals, a sequence of sets is minimizing if and only if the geometric functional $\beta(\Omega)$ tends to zero. This provides a complete characterization of the optimizing sequences in terms of a purely geometric quantity.

B. Quantitative Inequalities (Theorems 1.2, 1.3, 1.4)

The authors establish sharp quantitative bounds relating the deficit of the functionals to the remainder term $\beta(\Omega)$ :

Pólya Torsion (Theorem 1.2):
$\frac{1}{23 \cdot 34 n^3} \beta(\Omega)^3 \le \frac{T(\Omega)P^2(\Omega)}{|\Omega|^3} - \frac{1}{3} \le \frac{n+1}{3} \beta(\Omega)$
- Significance: Improves upon previous results by providing a lower bound with the sharp exponent $\beta^3$ in dimension $n$ .
Pólya Eigenvalue (Theorem 1.3):
$\frac{\pi^2}{25 \cdot 34 \cdot n^3(2n-1)} \beta(\Omega)^4 \le \frac{\pi^2}{4} - \frac{\lambda(\Omega)|\Omega|^2}{P^2(\Omega)} \le \frac{\pi^2}{2} \beta(\Omega)$
- Significance: Establishes a lower bound with exponent 4, which is shown to be sharp.
Makai Functional (Theorem 1.4):
$\frac{n+1}{3n(2n-1)} \beta(\Omega) < \frac{1}{3} - \frac{T(\Omega)}{R_\Omega^2 |\Omega|} \le \frac{2}{3} \beta(\Omega)$
- Significance: The lower and upper bounds have the same linear exponent in $\beta(\Omega)$ , fully characterizing the convergence rate.

C. Results on $\alpha(\Omega)$ (Proposition 1.5)

The paper addresses the conjecture regarding the Pólya torsion functional's dependence on the width-to-diameter ratio $\alpha(\Omega)$ .

Result: It is proven that $\frac{T(\Omega)P^2(\Omega)}{|\Omega|^3} - \frac{1}{3} \ge C_1(n) \alpha(\Omega)$ and $\frac{\pi^2}{4} - \frac{\lambda(\Omega)|\Omega|^2}{P^2(\Omega)} \ge C_2(n) \alpha(\Omega)$ .
Sharpness: The exponent 1 for $\alpha(\Omega)$ is sharp.
Limitation: The paper proves that reverse inequalities (bounding $\beta$ or the functional deficit from above by $\alpha$ ) are false, using collapsing pyramids as counterexamples. This highlights that $\alpha$ is a coarser measure than $\beta$ for these specific problems.

D. Hersch Functional (Corollary 1.6)

A new upper bound is derived for the Hersch functional deficit:
$\lambda(\Omega)R_\Omega^2 - \frac{\pi^2}{4} \le \frac{\pi^2(n+1)}{4} \beta(\Omega)$
This removes the dependency on the eigenvalue of a lower-dimensional projection found in previous literature, at the cost of a dimension-dependent constant.

4. Significance and Impact

Resolution of Non-Existence of Optima: The paper provides a rigorous framework for analyzing shape optimization problems where the optimum does not exist in the standard class of sets. By quantifying the "distance" to the infimum via geometric remainders, it characterizes the asymptotic behavior of minimizing sequences.
Refinement of Geometric Measures: The distinction between $\alpha(\Omega)$ (width-based) and $\beta(\Omega)$ (perimeter/inradius-based) is crucial. The authors demonstrate that $\beta(\Omega)$ is the correct metric for these specific functionals, as $\alpha(\Omega)$ is insufficient to capture the necessary geometric constraints (e.g., collapsing pyramids have $\alpha \to 0$ but do not minimize the functionals in the same way as flattening cylinders).
Sharp Exponents: The paper determines the exact power laws (exponents) governing the convergence of these functionals to their limits. This is a significant advancement over qualitative inequalities, providing precise rates of convergence.
Methodological Advancement: The use of specific level sets of the distance function (specifically at $t = |\Omega|/P(\Omega)$ ) and the integration of concavity properties of the perimeter of inner parallel sets offers a powerful new technique for deriving quantitative spectral and torsional inequalities.

In summary, this work bridges the gap between qualitative shape inequalities and quantitative stability analysis for functionals where the optimal shape is a degenerate limit, offering precise geometric characterizations of the minimizing sequences.