🔢 mathematics

On the Brunn-Minkowski inequality for $q$ -th dual quermassintegrals with $q>n$

This paper resolves the Brunn-Minkowski inequality for $q$ -th dual quermassintegrals with $q>n$ by demonstrating its failure for general and certain symmetric convex bodies while establishing its validity for origin-symmetric bodies at the endpoint $q=n+2$ and for unconditional bodies throughout the range $0<q\le n+1$ , leading to new uniqueness results for dual curvature measures.

Original authors: Haizhong Li, Yao Wan, Yijia Zhang

Published 2026-08-05

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Haizhong Li, Yao Wan, Yijia Zhang

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 the world of shapes not as static drawings, but as living, breathing entities that can stretch, squish, and merge. In the branch of mathematics called convex geometry, scientists study "convex bodies"—think of these as perfectly smooth, bulging shapes like a basketball, a loaf of bread, or a polished stone, where if you draw a line between any two points inside, the whole line stays inside. For over a century, mathematicians have been obsessed with a golden rule called the Brunn-Minkowski inequality. It's like a cosmic law of volume: if you take two of these shapes and smash them together (a process called Minkowski addition), the resulting shape is always "fatter" than you'd expect if you just added their sizes up. It's the reason why a pile of sand is harder to push than a single grain, and it helps engineers design stronger bridges and physicists understand how matter behaves.

But there's a twist. What if we don't just measure the total volume of these shapes? What if we weigh them differently, giving more importance to the parts far from the center, or the parts close to the center? This is where dual quermassintegrals come in. Think of them as "flavored volumes." Instead of just counting how much space a shape takes up, we count it with a special recipe that changes based on a number called $q$ . If $q$ is small, the recipe is gentle; if $q$ is huge, the recipe goes wild, screaming about the edges and corners. For a long time, mathematicians knew this "flavored volume" rule worked perfectly when the recipe number $q$ was small (specifically, $q \le n$ , where $n$ is the number of dimensions). But what happens when we crank the dial up? What if we use a super-intense recipe where $q$ is bigger than the number of dimensions? Does the golden rule still hold, or does the universe of shapes break?

This is the puzzle that Li, Wan, and Zhang tackled in their new paper. They asked: Does the Brunn-Minkowski inequality still work when the flavor parameter $q$ is larger than the dimension of the space ( $q > n$ )?

The answer, it turns out, is a dramatic "It depends, and mostly no."

First, the authors showed that if you let the shapes be any old convex body (even a slightly lopsided one), the rule breaks completely as soon as $q$ gets bigger than $n$ . They proved this by taking a perfect sphere and giving it a tiny, wobbly wiggle. When they checked the math, the "flavored volume" of the wobbly sphere didn't behave nicely; it acted like a convex curve instead of a concave one, meaning the inequality flipped upside down. The "golden rule" was shattered.

But what if we restrict ourselves to perfectly symmetrical shapes, like a cube or a sphere that looks the same from every angle? Maybe the rule survives there? The authors dug deeper and found that even for these super-symmetrical shapes, the rule still fails if $q$ gets too high—specifically, if $q$ is bigger than $n + 2$ . They used a clever trick called "dimension reduction," imagining a 3D object as a flat rectangle squashed into a thin line. By doing this, they turned a complex 3D problem into a simple 2D puzzle involving rectangles. They calculated that for these rectangles, if the flavor number $Q$ (which relates to $q$ ) is greater than 4, the inequality fails. Translating this back to the real world, this means the rule breaks for any symmetrical shape if $q > n + 2$ .

However, the story doesn't end in failure. The authors found a "sweet spot" where the rule does hold true.

The Edge Case: They proved that for perfectly symmetrical shapes, the rule works exactly at the breaking point: when $q = n + 2$ . They did this by connecting the problem to an old, classic physics concept called the "polar moment of inertia" (which measures how hard it is to spin a shape). A century-old inequality by Hadwiger about spinning objects saved the day, proving the rule holds for this specific number.
The Special Club: They also found a special group of shapes called "unconditional convex bodies." These are shapes that are perfectly symmetrical across every coordinate axis (like a dice or a box aligned with the grid). For these shapes, the rule works for a much wider range: from $q = 0$ all the way up to $q = n + 1$ . To prove this, they used some heavy-duty mathematical tools called "Reilly formulas" and "Hardy inequalities," which are like advanced stress-tests for shapes, checking how they hold up under the weight of these weird "flavored" measurements.

So, what's the takeaway? The idea that the Brunn-Minkowski inequality works for all shapes with any flavor number $q$ is false. The universe of shapes is more fragile than we thought. If you crank the flavor dial too high ( $q > n$ ), the rule collapses for general shapes, and even for symmetrical ones, it collapses if you go past $n + 2$ . But, if you stick to the "unconditional" shapes (the grid-aligned ones) or hit the exact edge case of $n + 2$ for symmetrical shapes, the rule survives.

This isn't just a game of numbers. By figuring out exactly where the rule holds and where it breaks, the authors also solved a mystery about uniqueness. They showed that if you know the "flavored volume" distribution of these special shapes, you can uniquely identify the shape itself (up to scaling). It's like saying, "If I tell you exactly how this shape weighs at every distance from the center, you can tell me exactly what the shape is." This helps mathematicians understand the deep, hidden connections between the geometry of shapes and the way they distribute their mass, proving that even in the abstract world of high-dimensional math, there are strict limits to how much a shape can stretch before the rules of the game change.

Technical Summary: The Brunn-Minkowski Inequality for $q$ -th Dual Quermassintegrals with $q > n$

1. Problem Statement and Context

The paper investigates the validity of the Brunn-Minkowski inequality for the $q$ -th dual quermassintegral, denoted $\widetilde{V}_q(K)$ , when the index $q$ exceeds the dimension $n$ . The dual quermassintegral is defined for a star body $K$ containing the origin as:
$\widetilde{V}_q(K) = \frac{1}{n} \int_{S^{n-1}} \rho_K(u)^q \, du = \frac{q}{n} \int_K |x|^{q-n} \, dx,$
where $\rho_K$ is the radial function. The central question, posed by Sadovsky and Zhang [23], is whether the inequality
$\widetilde{V}_q(K+L)^{1/q} \geq \widetilde{V}_q(K)^{1/q} + \widetilde{V}_q(L)^{1/q}$
holds for $q > n$ under various symmetry assumptions (arbitrary, origin-symmetric, or unconditional).

While the inequality is known to hold for $0 < q \leq n$ (with the case $q=n$ being the classical Brunn-Minkowski inequality), the behavior for $q > n$ remained open. Previous work by Xu and Yang [28] established failure for $q > 2n+1$ in the origin-symmetric class. This paper aims to determine the sharp thresholds for failure and establish positive results in specific geometric classes.

2. Methodology

The authors employ a combination of variational analysis, dimension reduction, and weighted geometric inequalities.

2.1. Variational Analysis and Second Variation

The core of the negative results relies on analyzing the local $1/q$ -concavity of the measure $\mu_q(K) = \int_K |x|^{q-n} dx$ . Using the weighted second variation formula (Lemma 2.1) derived from Kolesnikov-Milman [17], the authors test the stability of the Euclidean ball $B^n_2$ under perturbations.

Arbitrary Bodies: They test odd perturbations (first spherical harmonics). The second variation is positive (indicating local convexity and thus failure of the inequality) if $q > n$ .
Origin-Symmetric Bodies: They test even perturbations. The local stability threshold is found to be $q \leq 2n+1$ , but this is shown to be insufficient for global validity.

2.2. Dimension Reduction

To construct global counterexamples for origin-symmetric bodies, the authors utilize a dimension reduction argument. They consider bodies of the form $K_\epsilon = A \times [-\epsilon, \epsilon]^{n-2}$ , where $A$ is a planar convex body. As $\epsilon \to 0$ , the $n$ -dimensional parameter $q$ effectively reduces to a planar parameter $Q = q - n + 2$ .

By analyzing planar rectangles, they show that the inequality fails if $Q > 4$ , which corresponds to $q > n + 2$ .
This construction bridges the gap between the local stability threshold and the global failure threshold.

2.3. Singular Weighted Reilly Formula and Hardy Inequalities

For the positive results, particularly for unconditional convex bodies, the authors develop a singular weighted Reilly formula (Proposition 6.5).

They solve a singular Neumann problem for the operator $L_\alpha = \Delta + \alpha \frac{x}{|x|^2} \cdot \nabla$ with weight $|x|^\alpha$ (where $\alpha = q-n$ ).
They derive asymptotic expansions for the solution near the origin (Lemma 6.3) to handle the singularity.
Crucially, they establish coordinate-slice Hardy-type inequalities (Lemmas 7.1 and 7.3) tailored to the sign of $\alpha$ . These inequalities control the unfavorable terms in the Reilly formula, allowing them to prove the non-negativity of a specific energy functional $I_\alpha(u)$ for $-n < \alpha \leq 1$ (i.e., $0 < q \leq n+1$ ).

3. Key Contributions and Results

3.1. Sharpness of the Range for Arbitrary Bodies

Theorem 1.3: The inequality fails for arbitrary convex bodies whenever $q > n$ .

Method: The authors construct smooth, strictly convex bodies by perturbing the Euclidean ball with a first spherical harmonic. The second variation of $\widetilde{V}_q(K)^{1/q}$ becomes positive, violating the required concavity.

3.2. Counterexamples for Origin-Symmetric Bodies

Theorem 1.4: The inequality fails for origin-symmetric convex bodies whenever $q > n + 2$ .

Method: Using the dimension reduction to planar rectangles, they show that for $Q = q - n + 2 > 4$ , the function $t \mapsto \widetilde{V}_q(K_t)^{1/q}$ is locally convex at $t=0$ , leading to a strict violation of the Brunn-Minkowski inequality. This improves upon the previous bound of $q > 2n+1$ .

3.3. Positive Result for the Endpoint $q = n + 2$

Theorem 1.5: For origin-symmetric convex bodies, the inequality holds at the endpoint $q = n + 2$ .

Method: The authors relate $\widetilde{V}_{n+2}(K)$ to the polar moment of inertia $I(K)$ via the identity $I(K) = \frac{n}{n+2} \widetilde{V}_{n+2}(K)$ for origin-symmetric bodies. They then invoke Hadwiger's inequality [10] for the polar moment of inertia, which states $I(K+L)^{1/(n+2)} \geq I(K)^{1/(n+2)} + I(L)^{1/(n+2)}$ .

3.4. Full Range for Unconditional Convex Bodies

Theorem 1.6: For unconditional convex bodies (invariant under coordinate reflections), the inequality holds for the full range $0 < q \leq n + 1$ .

Method: This is the paper's primary technical achievement. By combining the singular weighted Reilly formula with coordinate-slice Hardy inequalities, they prove the local $1/q$ -concavity of the measure $\widetilde{V}_q$ for unconditional bodies.
Equality Case: If $K, L$ are of class $C^2_+$ , equality holds if and only if $K$ and $L$ are dilates of each other.

3.5. Uniqueness of Dual Curvature Measures

As an application, the authors derive uniqueness results for the $(1, q)$ -th dual curvature measure $\widetilde{C}_{1,q}$ .

Corollary 8.4: For unconditional $C^2_+$ bodies, if $\widetilde{C}_{1,q}(K, \cdot) = \widetilde{C}_{1,q}(L, \cdot)$ with $0 < q \leq n+1$ , then $K=L$ (unless $q=1$ , where they are dilates).
Corollary 8.5: For origin-symmetric $C^2_+$ bodies, if $\widetilde{C}_{1,n+2}(K, \cdot) = \widetilde{C}_{1,n+2}(L, \cdot)$ , then $K=L$ .

4. Significance and Claims

The paper claims to resolve Problem 1.2 posed by Sadovsky and Zhang [23] by establishing the precise boundaries of the Brunn-Minkowski inequality for dual quermassintegrals in the regime $q > n$ .

Refutation of Global Validity: It demonstrates that the inequality does not extend to $q > n$ for general bodies, nor to $q > n+2$ for origin-symmetric bodies, effectively narrowing the open range for origin-symmetric bodies to $n < q \leq n+2$ .
Conjecture: The authors propose Conjecture 1, suggesting that the inequality might hold for origin-symmetric bodies in the remaining gap $n < q < n+2$ .
Unconditional Case: The paper provides a definitive positive answer for unconditional bodies, extending the valid range up to $q = n+1$ . This is achieved through a novel adaptation of the Reilly formula to singular weights, a technique inspired by Hu-Ivaki [12] but extended to cover the full range $0 < q \leq n+1$ .
Applications: The results immediately yield Minkowski-type inequalities and uniqueness theorems for dual curvature measures in the specified classes, addressing Problem 9 of Sadovsky and Zhang regarding the equality case.

The work highlights the delicate interplay between the index $q$ , the symmetry of the bodies, and the geometric structure (unconditional vs. general), showing that while symmetry helps, it is not sufficient to preserve the inequality for all $q > n$ .

Technical Summary: The Brunn-Minkowski Inequality for qqq-th Dual Quermassintegrals with q>nq > nq>n