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On convex bodies with constant non-central sections

This paper proves that a symmetric convex body of revolution in $\mathbb{R}^4$ containing the unit ball must be a Euclidean ball if its hyperplane sections tangent to the unit ball have constant area $A$ , provided $A$ satisfies specific arithmetic conditions related to continued fractions that hold for a set of positive Hausdorff dimension.

Original authors: J. Haddad, D. Ryabogin

Published 2026-05-04

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CC BY 4.0

Original authors: J. Haddad, D. Ryabogin

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 have a mysterious, perfectly symmetrical 4-dimensional object (a "convex body") floating in space. Inside this object, there is a perfect, standard-sized 4D ball (like a 4D orange).

Now, imagine you have a magical 3D knife that can slice through this object. You are only allowed to make cuts that just barely touch the surface of the inner 4D ball (tangent cuts).

The Big Question:
If you slice this object from every possible angle using this magical knife, and every single slice you get has the exact same surface area, does that mean your mysterious object is actually just a perfect 4D ball itself?

For a long time, mathematicians didn't know the answer. It's like asking: "If I cut a loaf of bread from every side and every slice is the same size, is the loaf perfectly round?" In most cases, the answer is "maybe," but in this specific 4D scenario, the authors of this paper say: "Yes, but only if the size of the slice is a very specific, 'weird' number."

Here is how they figured it out, broken down into simple concepts:

1. The Shape of the Mystery

The authors focused on a special type of object called a "body of revolution." Think of a vase or a spinning top. If you spin a shape around an axis, you get a 3D object. In this paper, they are spinning a 2D shape around an axis to create a 4D object. This symmetry makes the math easier to handle.

2. The "Magic Number" of the Slice

The key to the proof is the area ( $A$ ) of the slice.

If the slice area is a "normal" number, the object might be a ball, or it might be a weird, squashed shape that just happens to have the same slice size.
However, the authors found that if the slice area corresponds to a specific "magic number," then the object must be a perfect ball.

3. The "Irrational Dance" (The Core Mechanism)

How did they find these magic numbers? They turned the geometry problem into a dance problem.

Imagine a point spinning around a circle.

Every time you make a slice, the point moves a specific angle.
The size of the slice determines how big that angle is.
The authors discovered that if the angle is "irrational" (meaning it never repeats a pattern exactly, like $\pi$ or $\sqrt{2}$ ), the point will eventually visit every part of the circle.
They proved that if the point visits every part of the circle without landing on a few "forbidden zones" (which represent the weird, non-ball shapes), then the object must be a ball.

4. The "Continued Fraction" Filter

This is where the math gets tricky, but here is the simple version:
To know if an angle is "safe" (i.e., it avoids the forbidden zones), you have to look at its continued fraction.

Think of a continued fraction as a recipe for a number. You write the number as a list of integers: $0; 2, 1, 3, 5, \dots$
The authors found that if the numbers in this recipe follow certain rules (like being even or odd in a specific pattern), the "dance" is safe.
If the recipe is "weird" enough (mathematically speaking, having a positive "Hausdorff dimension," which is a fancy way of saying there are infinitely many of these numbers and they are spread out in a complex way), then the object is guaranteed to be a ball.

5. What They Actually Proved

The paper does not say that every slice size proves the object is a ball.

The Claim: They proved that there is an infinite set of slice sizes (areas) for which the answer is definitely "Yes, it's a ball."
The Catch: These sizes are determined by the "recipe" (continued fraction) of the angle. If the recipe is "too simple" or "too regular," the proof doesn't work. But if the recipe is sufficiently complex and follows their specific rules, the mystery object is a ball.
The Result: They showed that this set of "magic slice sizes" is not empty; in fact, it's a huge, complex set that is mathematically "large" (positive Hausdorff dimension).

Summary Analogy

Imagine you are trying to identify a suspect in a lineup.

The Suspect: A 4D shape.
The Evidence: The size of the shadow (slice) it casts from every angle.
The Detective's Rule: "If the shadow size is a 'normal' number, the suspect could be anyone. But if the shadow size is a 'weird' number (one with a complex, non-repeating recipe), then the suspect is definitely the perfect ball."

The authors didn't solve the case for every possible shadow size, but they found a massive, infinite collection of "weird" shadow sizes where the perfect ball is the only possible answer. They also showed that these "weird" numbers are common enough to form a significant mathematical structure.

1. Problem Statement

The paper addresses a fundamental question in geometric tomography known as the Barker–Larman conjecture. The conjecture asks whether a convex body $C_1$ in $\mathbb{R}^n$ is uniquely determined by the volumes of its sections by hyperplanes tangent to the unit Euclidean ball $B_n$ , provided $C_1$ contains $B_n$ .

Specifically, if $C_1$ and $C_2$ are convex bodies containing $B_n$ , and for every hyperplane $H$ tangent to $B_n$ , the $(n-1)$ -dimensional volume of the sections is equal ( $|C_1 \cap H|_{n-1} = |C_2 \cap H|_{n-1}$ ), does it follow that $C_1 = C_2$ ?

While the problem is solved for central sections (hyperplanes passing through the origin) via the Radon transform, the non-central case remains largely open. The authors focus on a specific, highly symmetric case:

Dimension: $n=4$ .
Symmetry: The body $C$ is a symmetric body of revolution around an axis.
Condition: The body contains the unit ball $B_4$ , and all sections by hyperplanes tangent to $B_4$ have a constant area $A$ .

The Core Question: If such a body $C$ has constant section area $A$ , must $C$ be a Euclidean ball?

2. Methodology

The authors employ a multi-stage approach combining convex geometry, calculus, and number theory (specifically Diophantine approximation and continued fractions).

A. Geometric Reduction

Dimensional Reduction: Since $C$ is a body of revolution in $\mathbb{R}^4$ , it is determined by its 2D profile $K$ in the plane $\langle e_1, e_2 \rangle$ . The problem reduces to determining if $K$ is a Euclidean disk.
Area Equation: The authors derive an integral equation relating the area $A$ of the 3D section to the profile function $f(x)$ of $K$ . By differentiating this equation with respect to the slope $s$ of the tangent line, they eliminate the function $f$ and derive a polynomial relationship involving the intersection points of the tangent line with the boundary $\partial K$ .
The "Next Point" Map: They define a map $T_K$ that takes a point $p$ on the boundary of $K$ and maps it to the "next" intersection point of the tangent line through $p$ with $\partial K$ . For a Euclidean ball $R B_2$ , this map is explicit. The goal is to show that if the section area is constant, $T_K$ must coincide with the map for a Euclidean ball.

B. Dynamical Systems and Irrational Rotations

The authors transform the geometric problem into a problem about irrational rotations on a circle.

They define a rotation number $\gamma_A = \frac{1}{\pi} \arctan\left( (A/\omega_3)^{1/3} \right)$ , where $\omega_3$ is the volume of the unit 3-ball.
They construct a sequence of points $u_k$ on the circle $S_R$ (radius $R$ ) generated by rotating by the angle $\pi \gamma_A$ .
The Shrinking Interval Argument: They define a sequence of "forbidden" intervals $I_k$ on the circle that shrink to a point as $k \to \infty$ . These intervals represent the range of angles where the geometric uniqueness of the "next point" fails (i.e., where the polynomial equation for the intersection point has multiple valid solutions).
Uniqueness Condition: They prove that if the orbit of the rotation $u_k$ never falls into the shrinking intervals $I_k$ , then the profile $K$ must be a Euclidean disk.

C. Arithmetic Analysis (Continued Fractions)

The condition that the orbit avoids the shrinking intervals is an arithmetic property of the rotation number $\gamma_A$ .

The authors analyze the continued fraction expansion of $\gamma_A = [a_0, a_1, a_2, \dots]$ .
They relate the convergence speed of the rotation (how close $u_k$ gets to the "danger zone") to the growth rate of the continued fraction coefficients $a_j$ .
They establish Theorem 14, which provides a finite set of inequalities involving the coefficients $a_j$ $a_{j}$ that guarantee the orbit avoids the forbidden intervals for all $k$ $k$ .
- Specifically, they require specific parity conditions on the coefficients (e.g., $a_1=2$ , $a_3$ odd, $a_k$ even for $k \ge 4$ ) to ensure the denominators of the convergents behave in a way that allows the induction to hold.

3. Key Contributions and Results

Main Theorem (Theorem 1)

The authors prove that there exists an infinite set $\mathcal{A} \subset \mathbb{R}$ of positive Hausdorff dimension such that if the constant section area $A \in \mathcal{A}$ , then any symmetric convex body of revolution in $\mathbb{R}^4$ containing $B_4$ with constant section area $A$ is necessarily a Euclidean ball.

The Set $\mathcal{A}$

Definition: $\mathcal{A}$ consists of values $A$ for which the associated rotation number $\gamma_A$ is irrational and satisfies the specific continued fraction inequalities derived in Theorem 14.
Size: The set $\mathcal{A}$ is uncountable and has positive Hausdorff dimension.
Explicit Examples: The paper provides explicit numerical examples (e.g., $A_0 \approx 81.58$ ) where the conditions are verified, proving the set is non-empty.

Technical Innovations

Reduction to Arithmetic: The paper successfully bridges a geometric uniqueness problem with the arithmetic properties of irrational numbers, specifically using the theory of continued fractions to control the density of orbits.
Handling Non-Central Sections: Unlike central sections where Fourier analysis works, non-central sections require this novel dynamical systems approach to handle the lack of symmetry in the projection.

4. Significance and Implications

Progress on Barker-Larman: This is a significant step toward resolving the Barker-Larman conjecture in dimension 4. While the general case remains open, this result proves uniqueness for a large class of symmetric bodies under specific arithmetic conditions on the section volume.
Methodological Bridge: The work demonstrates a powerful new technique for geometric tomography: translating geometric constraints into Diophantine approximation problems. This suggests that "freak" parameters (like those in Ungar's theorem mentioned in the introduction) play a crucial role in uniqueness problems.
Hausdorff Dimension vs. Measure: The authors show the set of valid parameters has positive Hausdorff dimension but leave open the question of whether it has positive Lebesgue measure (Question 15). This highlights the delicate nature of the arithmetic conditions required for uniqueness.
Symmetry Assumption: The authors note that the symmetry assumption is likely not essential for the result but simplifies the proof significantly. Removing it would be a major future challenge.

Conclusion

Haddad and Ryabogin provide a rigorous proof that in $\mathbb{R}^4$ , a symmetric body of revolution with constant non-central section area is a Euclidean ball, provided the area value satisfies specific arithmetic conditions related to the continued fraction expansion of a derived rotation number. This establishes the existence of a "large" (in terms of dimension) set of such values, moving the Barker-Larman conjecture from a purely geometric question to one deeply intertwined with number theory.