The Makai inequality in higher dimensions: qualitative and quantitative aspects

1. Problem Statement

The paper investigates the behavior of a specific scaling-invariant functional defined on the class of non-empty, open, bounded, and convex subsets of $\mathbb{R}^n$ (denoted as $\mathcal{K}_n$). The functional $F(\Omega)$ combines the torsional rigidity of the Laplacian, the perimeter, and the Lebesgue measure (volume) of the domain:

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

Where:

• $T(\Omega) = \max_{\phi \in H^1_0(\Omega), \phi \not\equiv 0} \frac{(\int_\Omega \phi dx)^2}{\int_\Omega |\nabla \phi|^2 dx}$ is the torsional rigidity.
• $P(\Omega)$ is the perimeter (Hausdorff measure of the boundary).
• $|\Omega|$ is the volume.

Context:

• Minimization: Pólya established a lower bound ($1/3$ in 2D, extended to higher dimensions) attained asymptotically by flattening cylinders.
• Maximization: Makai proved a sharp upper bound of $2/3$ for $n=2$, attained asymptotically by flattening triangles.
• The Open Problem: In [7], it was conjectured that for arbitrary dimension $n \ge 2$, the sharp upper bound is:
  $$\sup_{\Omega \in \mathcal{K}_n} F(\Omega) = \frac{2n^2}{(n+1)(n+2)}$$
  This conjecture was supported by the observation that sequences of flattening cones approach this value.

The paper aims to:

1. Prove this conjecture (The Makai inequality in $\mathbb{R}^n$).
2. Determine if the supremum is attained by a specific domain or only by a sequence.
3. Provide quantitative stability estimates relating the deficit of $F(\Omega)$ from the optimal constant to geometric parameters describing the "thinness" and shape of the domain.

---

2. Methodology

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

A. Geometric Preliminaries

• Distance Function: They utilize the distance function $d(x, \partial \Omega)$ from the boundary. Due to the convexity of $\Omega$, this function is concave.
• Inner Parallel Sets: They define $\Omega_t = \{x \in \Omega : d(x, \partial \Omega) > t\}$ and use the coarea formula to relate the volume and perimeter of these sets to the distance function.
• Support Function & Width: They introduce the minimal width $w_\Omega$ and diameter $\text{diam}(\Omega)$ to define the thinness parameter $\alpha(\Omega) = w_\Omega / \text{diam}(\Omega)$.
• Tangential Bodies: A key concept is the "tangential body," where inner parallel sets are homothetic to the original set ($\Omega_t = (1 - t/R_\Omega)\Omega$).

B. Upper Bound Derivation (Proof of Theorem 1.1)

1. Torsion Estimate: They use the inequality $T(\Omega) \le \int_\Omega d(x, \partial \Omega)^2 dx$.
2. Concavity Argument: Let $L(t) = P(\Omega_t)^{1/(n-1)}$. By the Brunn-Minkowski inequality, $L(t)$ is concave.
3. Linear Approximation: They construct a linear function $\lambda(t)$ such that $\int_0^{R_\Omega} \lambda(t)^{n-1} dt = |\Omega|$.
4. Integral Comparison: Using the concavity of $L(t)$ and the properties of $\lambda(t)$, they prove that $\int_0^{R_\Omega} L(t)^{n-1} t^2 dt \le \int_0^{R_\Omega} \lambda(t)^{n-1} t^2 dt$.
5. Optimization: The problem reduces to maximizing the integral of the linear function subject to the volume constraint. This leads to an algebraic optimization problem involving a parameter $y \in [0,1]$.
6. Algebraic Verification: They define a polynomial function $g(z)$ (where $z = 1-y$) and prove via derivative analysis that $g(z) \le 0$ for all $z \in [0,1]$, thereby establishing the upper bound.

C. Quantitative Analysis (Proof of Theorem 1.3)

To characterize the optimizing sequences, they introduce a new remainder term:
$$\gamma(\Omega) = n - \frac{P(\Omega)R_\Omega}{|\Omega|}$$
They analyze the deficit $C_{opt} - F(\Omega)$ by expanding the integral estimates near the optimal case (where $z \to 1$). They show that the deficit is bounded below by a power of $\gamma(\Omega)$.

---

3. Key Contributions and Results

Theorem 1.1: The Sharp Makai Inequality

For any $\Omega \in \mathcal{K}_n$ with $n \ge 2$:
$$F(\Omega) \le \frac{2n^2}{(n+1)(n+2)}$$

• Sharpness: The constant is sharp. It is asymptotically attained by a sequence of flattening cones (domains where the height tends to zero while the base remains fixed).
• Significance: This confirms the conjecture from [7] and generalizes Makai's 2D result to arbitrary dimensions.

Theorem 1.2: Non-Attainment and Thinness

The supremum is not attained by any single convex set $\Omega \in \mathcal{K}_n$.

• If $\{ \Omega_k \}$ is a maximizing sequence (i.e., $F(\Omega_k) \to \text{sup}$), then the thinness parameter $\alpha(\Omega_k) = \frac{w_{\Omega_k}}{\text{diam}(\Omega_k)}$ must converge to 0.
• Implication: The maximizing sequence must consist of "thinning" domains. Since a set with $\alpha(\Omega)=0$ would have an empty interior (not in $\mathcal{K}_n$), the maximum is never reached.

Theorem 1.3: Quantitative Stability

The paper provides a quantitative estimate linking the functional deficit to the geometric parameter $\gamma(\Omega)$:
$$\frac{2n^2}{(n+1)(n+2)} - F(\Omega) \ge C_1(n) \cdot \gamma(\Omega)^n$$
Where $C_1(n)$ is an explicit constant depending on $n$.

• Interpretation: $\gamma(\Omega)$ measures the deviation of $\Omega$ from being a tangential body.
• Corollary 1.4: A maximizing sequence for $F(\Omega)$ is characterized as a sequence of flattening tangential bodies. This means the domains must simultaneously become infinitely thin ($\alpha \to 0$) and geometrically approach the shape of a tangential body ($\gamma \to 0$).

---

4. Significance and Impact

1. Resolution of a Conjecture: The paper definitively solves the open problem regarding the sharp upper bound of the torsional rigidity functional in higher dimensions, confirming the specific value $\frac{2n^2}{(n+1)(n+2)}$.
2. Geometric Characterization: It moves beyond qualitative bounds to provide a quantitative stability result. By introducing $\gamma(\Omega)$, the authors offer a precise metric to describe how close a domain is to the optimal shape, distinguishing between different types of "thin" domains (e.g., flattening cones vs. flattening cylinders).
3. Methodological Advancement: The proof technique, utilizing the concavity of the perimeter function of inner parallel sets combined with a linear approximation and algebraic polynomial analysis, offers a robust framework for similar isoperimetric-type problems involving torsional rigidity.
4. Distinction from Minimization: The paper highlights the contrast between the minimization problem (Pólya's inequality, attained by flattening cylinders) and the maximization problem (Makai's inequality, attained by flattening cones), showing that the geometric requirements for optimality are fundamentally different.

In summary, this work provides a complete qualitative and quantitative description of the Makai inequality in $\mathbb{R}^n$, establishing the optimal constant, proving non-attainment, and characterizing the asymptotic behavior of optimizing sequences through novel geometric parameters.