Convex bodies with equipotential circles
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.