Unit lemniscates contained in the unit ball

Abstract:

Let $\{ {A_1},{A_2}, \ldots ,{A_\nu }\} \equiv A$ be a set of points in ${E^n}$. Let $E(A)$ be the set of points in ${E^n}$ such that $\Pi _{k = 1}^\nu \overline {p{A_k}} \leqslant 1$ (where $\overline {p{A_k}}$ denotes the Euclidean distance between $p$ and ${A_k}$), and call this set the unit lemniscate with focal set $A$. It is shown that if the vertices of a regular tetrahedron lie at the distance $\delta \in (0,1)$ from the origin, then they are the foci of a unit lemniscate contained in the open unit ball of ${E^3}$ if and only if the sign of $36 - 64\delta + 2{\delta ^2} - 64{\delta ^3} + 36{\delta ^4} + 27{\delta ^6}$ is positive.