Convex bodies with equipotential circles

TL;DR

This work introduces and analyzes equipotential and equireciprocal structures inside convex bodies to obtain rigidity results. In the plane, it proves that a convex body $K\subset\mathbb{R}^2$ containing an interior equipotential circle $\mathcal B$ with centre $O$ must be centrally symmetric about $O$, and becomes a disc if no tangent chord to $\mathcal B$ subtends a $\pi/2$ angle at $O$; the proof combines an invariant-measure approach with Denjoy theory and Barker–Larman results. In 3D, the authors show that if every $2$-D section through a fixed interior point has an equipotential circle, then the body is an ellipsoid, and they apply a false-centre framework to deduce sphere/ellipsoid characterizations. They also extend the notions to equireciprocal circles and spheres, proving analogous rigidity: an interior equireciprocal circle or sphere forces the body to be a disc or ball, respectively. The paper further outlines open problems and lays groundwork for higher-dimensional generalizations, highlighting new rigidity phenomena for convex bodies under equipotential constraints.

Abstract

Given a convex body $K\subset \mathbb R^2$ we say that a circle $Ω\subset \text{int} \ K$ is an equipotential circle if every tangent line of $Ω$ cuts a chord $AB$ in $K$ such that for the contact point $P=Ω\cap AB$ it holds that $|AP|\cdot|PB|=λ$, for a suitable constant number $λ$. The main result in this article is the following: Let $K\subset\mathbb R^2$ be a convex body which has an equipotential circle $\mathcal B$ with centre $O$ in its interior. Then $K$ has centre of symmetry at $O$, moreover, if none chord of $K$ which is tangent to $\mathcal B$ subtends an angle $π/2$ from $O$, then $K$ is a disc. We also derive some results which characterizes the ellipsoid and the sphere in $\mathbb R^3$ and introduce also the concept of equireciprocal disc.

Figures (11)

• Figure 1: The Limaçon of Pascal has an equichordal point
• Figure 2: A curve with an equipotential point $O$
• Figure 3: Ellipses have equipotential circles
• Figure 4: Triangles $\triangle AA'P$ and $\triangle BB'P$ are similar
• Figure 5: It holds that $|A_1P|\cdot|PA_0|=\tan \theta \cdot \tan (\alpha-\theta)$

Theorems & Definitions (17)

• Definition 1
• Theorem 1
• Lemma 1
• Lemma 2
• Remark 1
• Example 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5