On a Question of Erdős on Lemniscates in R^m: Connected Lemniscates Lie in a Ball of Radius 2

For points \(x_1,\dots,x_n\) in \(\mathbb{R}^m\) let \(E=\{x\in\mathbb{R}^m:\prod_{i=1}^n|x-x_i|\le1\}\). A question of Erdős, posed in Hayman's collection of problems of 1974 and recorded as Problem 4.23 by Hayman and Lingham, asks how large the volume of \(E\) can be and whether a connected \(E\) can be covered by a ball of radius \(2\); for \(m=2\) the covering statement is a theorem of Pommerenke (1959). We answer the covering question: if \(E\) is connected, then \(|x-c|<2\) for every \(x\in E\), where \(c\) is the centroid of \(x_1,\dots,x_n\), for all \(m\) and \(n\). Projecting \(E\) onto the line through \(c\) and a point of \(E\) reduces the question to Pommerenke's planar theorem for a polynomial with real zeros; we prove that theorem again, in the closed-set form needed. Collinear Chebyshev configurations give connected sets \(E\) with \(\max_{x\in E}|x-c|=2\cdot2^{-1/n}\), so the constant \(2\) cannot be lowered uniformly in \(n\), already for \(m=1\). The same projection shows that for each \(n\) the best radius about the centroid is the same in every dimension; it equals \(2\cdot2^{-1/n}\) for \(n\le2\), and whether it does for all \(n\), as numerical experiments suggest, is open, as is the volume question. The question is Erdős's and the planar case is Pommerenke's; the note contributes the reduction, which gives the covering statement in every dimension and the dimension-independence of the best radius about the centroid. We found no proof of the covering statement for \(m\ge3\) in the sources we searched. This is an unrefereed note.