Convex bodies with equipotential circles

TL;DR

This paper characterizes convex bodies with equipotential circles, proving symmetry and disc conditions, and extends results to ellipsoids and spheres, introducing the concept of equireciprocal discs.

Contribution

It provides a geometric characterization of convex bodies with equipotential circles, establishing symmetry and specific shape conditions, and introduces the concept of equireciprocal discs.

Findings

Convex bodies with an interior equipotential circle are centrally symmetric.

Such bodies are disks if no tangent chord subtends a right angle from the center.

The paper extends characterizations to ellipsoids and spheres in higher dimensions.

Abstract

Given a convex body $K\subset \mathbb R^2$ we say that a circle $\Omega\subset \text{int} \ K$ is an equipotential circle if every tangent line of $\Omega$ cuts a chord $AB$ in $K$ such that for the contact point $P=\Omega\cap AB$ it holds that $|AP|\cdot|PB|=\lambda$, for a suitable constant number $\lambda$. 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 $\pi/2$ 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.