A problem on a geometric property of lemniscates

Abstract:

Let ${R^3}$ be the Eulcidean space and let ${p_n}$ be the product defined by ${p_n}(W,{W_k}) = \Pi _{k = 1}^n\left | {W - {W_k}} \right |$, $W$, ${W_k} \in {R^3}$, where $\left | {W - {W_k}} \right |$ is the distance between $W$ and ${W_k}$. Let $C(n)$ be the class of all such products with the same degree $n$. For any product $p$, we call $E(p) = \{ W:p(W) \leqslant 1\}$ the lemniscate of $p$. We recently proved that if ${p_n}(W,{W_k})$ and $p_n^*(W,W_k^*)$ are two products in $C(n)$ such that $E({p_n}) \subseteq E(p_n^*)$, and if all zeros ${W_k}$ of ${p_n}$ lie on the same plane, then ${p_n} \equiv p_n^*$. We then asked whether this result is sharp. In this note, we answer this question in the affirmative sense.