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A Sharp Diameter–Area Rhombus Inclusion and Applications to Newtonian Capacity

This paper establishes a sharp geometric inclusion theorem showing that the Steiner symmetral of a planar convex body contains a canonical rhombus with specific diagonal dimensions, which is then utilized to derive a scale-invariant inequality for Newtonian capacity.

Original authors: Juan Ignacio Guzmán

Published 2026-09-09

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Juan Ignacio Guzmán

Original paper licensed under CC BY 4.0 (https://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

In the world of physics, some objects are better at holding an electric charge than others. Imagine a metal plate sitting in space; if you try to put electricity on it, the charge spreads out across the surface. The ease with which this happens is called capacity. A large, flat disk holds a lot of charge, while a thin, needle-like shape holds very little. Scientists have long been interested in the relationship between this capacity and the simple geometric shapes of these objects. They want to know: if you know the size and the shape of a conductor, can you predict exactly how much charge it can hold? This question becomes particularly tricky when the object is very flat, like a coin, or very long and thin, like a wire. Understanding these limits helps mathematicians and physicists describe the fundamental rules that govern how energy and matter interact in space.

A recent study by Juan Ignacio Guzmán tackles this problem by focusing on flat, convex shapes—objects that have no dents or holes, like a smooth stone or a stretched rubber band. The researcher wanted to find a precise rule connecting the area of such a shape, its longest possible width (diameter), and its ability to hold a charge. The core of the discovery is a geometric insight: no matter how oddly shaped a flat, convex object is, if you rearrange its parts to be perfectly balanced around its longest line, the resulting shape will always contain a specific, diamond-like figure inside it. This hidden diamond is not just a random shape; its dimensions are strictly determined by the original object's area and its longest width. The researcher proved that this diamond is the largest possible one that can be guaranteed to fit inside any such rearranged shape, making the rule as tight and precise as mathematics allows.

Once this hidden diamond was identified, the study used it as a stepping stone to calculate the capacity of the original object. By fitting a series of smooth, oval shapes inside this diamond and finding the best possible fit, the author derived a new, sharper formula. This formula works especially well for objects that are very long and thin compared to their width. In these extreme cases, the new rule provides the most accurate prediction possible for how the capacity behaves, confirming that the mathematical behavior of these slender shapes follows a specific, predictable pattern. The study also showed that while simple shapes like circles are often the most efficient, the new rule is necessary to understand the full range of possibilities, as no single simple formula can describe every shape perfectly.

The power of this discovery extends beyond flat plates into the three-dimensional world. The researcher applied this new two-dimensional rule to solid objects, such as a block of metal or a sphere, by looking at their thinnest cross-sections. By combining the new flat-shape rule with other known geometric facts about how solid objects project their shadows, the study produced a stronger guarantee for the capacity of any solid object. Previously, scientists knew that the capacity of a solid object could not fall below a certain value relative to its surface area, but that limit was somewhat loose. The new work tightens this limit significantly, proving that the capacity is always at least a specific, higher fraction of what was previously thought to be the minimum. This improvement is not a small adjustment; it raises the guaranteed lower bound from a value of three-quarters to nineteen-twenty-fifths.

This result is significant because it holds true for any solid object, no matter how jagged or irregular its surface might be, and without needing to assume the object is perfectly smooth. The proof is rigorous and relies on pure logic rather than computer simulations or approximations. It confirms that the relationship between an object's size, its surface area, and its ability to hold an electric charge is more constrained than previously understood. By finding this sharper boundary, the study closes a gap in our understanding of how geometry dictates physical properties, offering a more precise map for how electricity behaves on the surfaces of the world around us.

Technical Summary: A Sharp Diameter–Area Rhombus Inclusion and Applications to Newtonian Capacity

Problem Statement
The paper addresses the interaction between electrostatic capacity (specifically Newtonian capacity in $\mathbb{R}^3$ and 1-Riesz capacity in $\mathbb{R}^2$ ) and elementary geometric data, namely area and diameter. A central open problem in this domain is the Pólya–Szegő conjecture, which posits a lower bound for the capacity of a compact convex conductor $K \subset \mathbb{R}^3$ in terms of its intrinsic surface area $S(K)$ . While the conjectured sharp bound involves a constant of $2\sqrt{2}/\pi \approx 0.9003$ , previous universal estimates had established a lower constant of $3/4$ . The paper seeks to improve this universal constant by deriving a sharp geometric inclusion for planar convex bodies (plates) and applying it to the three-dimensional case.

Methodology
The author employs a combination of geometric analysis, potential theory, and optimization techniques:

Steiner Symmetrization and Geometric Inclusion: The core geometric tool is Steiner symmetrization about a diametral axis. The author proves that for any planar convex body $P$ with area $A$ and diameter $D$ , its Steiner symmetral contains a "canonical rhombus" with diagonals $D$ and $A/D$ . This inclusion is proven to be sharp via a degenerating family of convex kites.
Optimization of Inscribed Ellipses: Within the canonical rhombus, the author considers the full family of coaxial ellipses. By optimizing the capacity of these inscribed ellipses (using the known formula for the capacity of an ellipse involving complete elliptic integrals), a sharp lower bound for the capacity of the original body is derived as a function of the ratio $A/D^2$ .
Asymptotic Analysis: The behavior of the capacity is analyzed in the "slender" regime (where $A/D^2 \to 0$ ) to determine the leading-order asymptotic constant.
Three-Dimensional Application: The planar results are extended to $\mathbb{R}^3$ $R^{3}$ by combining the plate inequality with:
- Minimum-width projection arguments.
- Volume estimates derived from concave fibers.
- A diameter–perimeter coupling for the projected shape.
- Xiao's existing volume-sensitive capacity inequality.

Key Contributions and Results

Sharp Hidden-Rhombus Theorem (Theorem 1.1): The paper establishes that the Steiner symmetral of a planar convex body $P$ (with area $A$ and diameter $D$ ) contains a rhombus with orthogonal diagonals $D$ and $A/D$ . The coefficient of $A/D$ is proven to be sharp (equal to 1), demonstrated by a family of kites where the area approaches zero relative to the diameter.
Optimized Diameter–Area Capacity Inequality (Theorem 1.2): Using the rhombus inclusion, the author derives a scale-invariant capacity inequality:
$\mathcal{C}(P) \geq \frac{D}{2} \phi\left(\frac{A}{D^2}\right)$
where $\mathcal{C}(P) = \text{Cap}(P)/(4\pi)$ and $\phi(\eta)$ is an explicit function defined by maximizing over a parameter $q$ involving complete elliptic integrals of the first kind. This bound retains both area and diameter information, unlike previous area-only estimates.
Slender-Profile Asymptotic (Theorem 1.3): The paper determines the sharp leading-order asymptotic for the capacity of slender convex plates. As the shape becomes highly eccentric ( $\eta = A/D^2 \to 0$ ), the capacity satisfies:
$\mathcal{C}(P) \sim \frac{D}{2 \log(1/\eta)}$
The constant $1/2$ in this asymptotic law is proven to be optimal.
Improved Universal 3D Constant (Theorem 1.4): By coupling the planar plate inequality with minimum-width geometry and volume estimates, the paper proves a new universal lower bound for the capacity of any compact convex conductor $K \subset \mathbb{R}^3$ with intrinsic surface area $S(K) > 0$ :
$\frac{\text{Cap}(K)}{2\sqrt{\pi S(K)}} \geq \frac{19}{25}$
This improves the previously known universal constant of $3/4$ (from Xiao's work) to $19/25 = 0.76$ . The proof relies on a case-by-case analysis of the projected isoperimetric ratio and diameter coupling, using exact rational arithmetic to verify the bounds.

Significance and Claims
The paper claims that the derived coefficient of $19/25$ is a rigorous improvement over the previous universal constant of $3/4$ without requiring assumptions of smoothness or strict convexity. The author emphasizes that the planar capacity bound itself is not claimed to be the absolute optimal for every fixed $A/D^2$ , but rather that the method of combining the sharp rhombus inclusion with optimized ellipses yields the best possible leading-order asymptotic for slender profiles and a strictly better universal constant in 3D.

The work isolates specific geometric losses in previous arguments, particularly in how the diameter and area interact in the capacity estimate. While the paper does not claim to solve the full Pólya–Szegő conjecture (which predicts a constant of $\approx 0.9003$ ), it narrows the gap and provides a new framework for analyzing capacity via diameter–area profiles. The author identifies several open problems, including the exact determination of the capacity profile $\mathcal{A}(\eta)$ for fixed eccentricity and the quantitative stability of the capacity near a disk.

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