Convex bodies with equipotential circles

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.